A month is not a particularly alarming amount of time for a chile pepper of any variety to stay green, especially depending on your soil and growing conditions. If the variety isn't used to the growing conditions, it's possible that this might also delay ripening.
Carolina Reapers are supposed to take between 70 to 90 days after the transplant before maturity, but other varieties may be earlier or later (it really depends on the variety). However, in some areas and/or soils (such as mine) it may take much, much longer for Carolina Reapers to mature (like maybe longer than the season). I believe the variety comes from South Carolina (so the growing conditions there may be more like ideal than many areas). I don't know what kind of soil they're used to (but probably the usual ideal for peppers).
I tried growing a Carolina Reaper this year, but I only got fruit indoors, while the Orange Carbonero chile plant I grew indoors next to it got loads of fruit after the transplant outside. I did, however, get a very large plant from the Carolina Reaper seeds (it may have needed a bigger container; I had it in a 5-gallon bucket, and it was bigger than all my other pepper plants). I also got plenty of fruit from a number of other varieties (Ring of Fire, Aji Omnicolor, Yatsufusa, Grandpa's Home, etc.) at varying stages of the season.
Peppers generally take between 60 and 120 days after the transplant to get mature fruit. (I've read about a whole lot of peppers, and that's the general range I've seen.) Most chile peppers take somewhat longer than sweet peppers to ripen, however, but there are early chile peppers. Ring of Fire is supposed to be 60 days, in fact (but, it was much later for me—although still productive and tasty).
I'm not sure about the Carolina Reaper, but many chile peppers (including all of those others I mentioned) will ripen a large amount of fruit at approximately the same time. So, I wouldn't be surprised if that happens. However, don't fret if it does. Chile peppers are normally pretty easy to dry, and if you dry them, you can use them at your leisure.
Here's a link that outlines some processes for drying peppers, including safety tips. It has good advice, and several methods, but some of what they say about the bowl method is rather overkill for some areas (although following all the advice certainly won't hurt). I mean, in my semi-arid area, I don't have to worry much about light levels when I dry peppers. I just wash them, dry them, put them in a bowl in the kitchen or dining room, and wait. I can even pile them higher than the stated one layer of peppers with good results (maybe three or more layers, but there is still a limit: if there are too many, the peppers may leak juice, attract fruit flies, and go bad; if they start to leak, you can separate them into more bowls). If your area isn't arid to semi-arid, you may want to worry about sunlight, ventilation, only doing one layer of peppers, etc. Ventilation seems to help things dry faster and stay good longer in my area, too; I recommend worrying about ventilation even if you don't have to, per se. It's just a good idea. I recommend using a dedicated fan, especially if you're drying them in a small room.
Chile peppers can take a long time to dry with the bowl method, but if all you care about is preserving them, it doesn't really matter (you can still use them while you're waiting for them to dry). If you're selling dry pepper pods on a schedule, it might matter to you, though.
If your peppers have thick walls, they may be more difficult to dry. You may have to cut them into strips (and in that case, you might want to use such as a dehydrator to dry them—but it can be done successfully just in the open air or in brown paper bags, in my area, at least). Food dehydrators may change the flavor of your peppers, however. I prefer just to let them sit and dry, for flavor.
Chile peppers (and peppers generally) are also extremely easy to freeze, however thick the walls are. Just wash them, put them in sandwich bags (or other zipper bags that you can freeze stuff in), and freeze them. If the peppers are whole, you shouldn't have to struggle to separate them when frozen. If they're cut into pieces, you may or may not, depending (but it shouldn't be too hard, most of the time).
I've personally never had peppers get mushy from being on the vine too long (unless they had blossom end rot), but it's possible in a more humid area, they might be prone to getting mushy (I don't know). So, if you're familiar with the concept of mushy peppers in your area, maybe someone else will answer with more details on how to handle the situation according to your needs. It's also possible that certain fertilizer programs may result in mushy peppers, but I don't know. Maybe they need extra silica or other nutrients to make them stronger (they might if they're container-grown with a pre-mixed soil). I would guess that disease might be responsible (but you'd think there would be visible signs of disease, too).
Carolina Reapers are supposed to turn red, eventually. I don't know of a kind that stays green. There is a new yellow variety, however. They may or may not turn red while drying them if it's not too cool, if you have to pick them green.
I do not recommend refrigerating your peppers. This will slow ripening, make drying them difficult (if not impossible) and limit their shelf life. (I'm actually kind of surprised they refrigerate fresh peppers in the supermarket. It doesn't seem to help, in my experience. Maybe it helps in humid areas.)
Direct sunlight might help your plants, but maybe not. It probably depends on your sun and air. If the UV index is high, it'll likely result in hotter peppers, though. I haven't noticed a big difference in ripening speed for chile peppers between partial shade and even more shade, personally (but I've only really tested it with Ring of Fire, a Capsicum annuum variety; the Carolina Reaper is Capsicum chinense).
Giving your plants more phosphorus could possibly speed ripening. I noticed the amount in your fertilizer is very low compared with the ratio in most NPK fertilizers. Phosphorus is often lower than nitrogen and potassium in fertilizers, but in this case, it's lower than usual. The nitrogen levels in that fertilizer are somewhat high, for peppers, I think (especially if it has already set fruit). Unless the soil is high in potassium, I might just give it a dose of monopotassium phosphate and see if it helps (but getting a soil test first would be a better idea, if you can do it).
I think fertilizing every 1-2 weeks is too often. The plant probably won't use it all, and eventually the nitrogen may burn your plants, cause root rot and stuff (and the soil may become too salty and maybe even too acidic). However, if the soil is high in nitrogen, I do recommend more sun (unless the UV rays are too harmful for the plants). If you've been fertilizing that often, you can (and probably should) take a break from fertilization (nitrogen especially) for a long time. Trying to balance out the nitrogen with other nutrients may help, but it may be dangerous if your soil has too many nutrients already.
High nitrogen (and/or low potassium) could possibly result in weak fruit (which may be the mushy fruit you mentioned). Weak plants/fruit may be more susceptible to disease. Disease can increase the spoilage rates, and even partially diseased fruits may be more difficult to dry successfully (you may not want to try it). If it's edible as is, you can freeze it, if you can't dry it. Silica and the proper proportion of calcium can also help strengthen a plant. Calcium can raise your soil pH, though. Silica sources include such as sand and rockdust.
