Hmmm. Where to start...
The soil you use in any garden is the number one most important part of the garden, so it worries me when you write that you're going to use bagged crap (yes, it'll be crap if it's from a box store) "topsoil" from Home Depot. This is, IMO, a terrible start to your garden.
Note - NEVER buy garden supplies at a box store. They are usually "faddish" (what the latest social media star is recommending, which is more often than not plain incorrect) and poorly sourced. Go to a local plant nursery or locally owned garden center, if possible. People there are always more knowledgeable than the people at box stores.
It also is a red flag to me that you're planning on lining the bottom of the bed with cardboard to keep out the Bermuda grass, even though you're planning on digging out the grass and its roots before building the bed. I am not familiar with that kind of grass, but as far as I know, no plant will willingly grow into an area where the sun isn't present. If you weren't digging out the roots I could see where you could have a potential problem, but with them gone, I don't think this is necessary.
And that's not even going into the horrible-ness of cardboard in gardens. I won't go into the details here (leave a comment if you want sources), but cardboard can be hydrophobic if it's waxed, which will cause rain to NOT drain at all, and contains potentially toxic inks. More importantly, it has been proven to kill ALL life in the soil unless it is thoroughly shredded. Given that you'd be putting it on the bottom of the bed, this is not much of concern unless you are growing deep-rooted plants, but never put cardboard on the top of your soil.
On the positive side of the ledger, I like that you're using cedar and that you're sealing it with a food-grade sealer. These are excellent ideas. I also like the hardware cloth on the bottom.
Here is what I would do in your shoes:
- Remove all Bermuda grass, as you've described.
- Build the bed as you've described, with the sealed cedar and a hardware cloth bottom.
- Your 16 sq. ft. x 18" deep bed will require about one cubic yard of topsoil (1 cubic yard will cover 100 sq ft. 3 inches deep. Your bed is roughly 100 sq. ft. in volume, unless my math is wrong - and it could be, so double-check it). Assuming that I'm correct, have about 3/4 yard of topsoil delivered from a reputable plant nursery. Get 1/4 yard compost, either from your own pile or a different source.
- Mix the compost and topsoil together, and add to the bed. Compact it a bit by gently pushing a garden rake or hoe onto it. You're aiming for a 2-3" space between the top of the soil and the top of your sides.
- Plant your crops.
- Mulch the crops after planting (or if you planted seeds, after they germinate). I prefer a single-season mulch like cocoa bean hulls, pine needles, rice hulls, marsh hay or whatever mulch is common to San Antonio. The mulch should be at least 1" deep (no more than 3/4" or so if you're using cocoa bean hulls)
- Water as needed. The mulch will both keep any weed seeds in the topsoil/compost mix from germinating and keep the soil from drying out quickly. As it breaks down, it'll feed your plants as well.
And that's all you need to do, except for some very occasional weeding and watering. At the end of the season, don't till the soil at all. Next year, just plant on top of the bed again and replace the mulch. I've been doing this for decades and it will work.
One other idea for you. If you're planning on mowing around the raised bed, it's a good idea to lay bricks horizontally (wide side up) or narrow pavers along all four sides of the bed. This gives the mower wheels room for travelling next to the bed and ensures that the mower cuts all of the grass, eliminating the need for using a string trimmer after mowing. It'll take some time to lay the bricks, but this is more than made up in the time you save not having to string-trim every week or so.
Off-topic, but I've been to San Antonio and loved the city. :)