The top and light green cucumber is the one in question. I thought it was fitting to use a cherry tomato for scale.
Is there anything wrong with letting a cucumber grow that large? Is there a well understood difference in flavor or texture? Any impact to the plant itself (aside from weight)?
I have heard that letting them get too large makes them sour/bitter.
According to: https://bonnieplants.com/growing/growing-cucumbers/ They say, "Don’t let the cucumbers get oversized or they will be bitter, and will also keep the vine from producing more."
No, there isn't. Providing you check them for flavor and make sure they haven't aged too far. At about 2 inches diameter. Larger than that and you'll start noticing flavor change and flesh hardening which is the important indicator that they've gone too far.
As long as the skin is soft, there is no flavor change and the flesh is fully moist, you will have a nice thin-slice cucumber for meal serving. We use a potato peeler to remove the skin, a sharp knife to cut them into 1/8-1/4 slices and either stack them on a plate or drop them in a bowl with a very light vinegar bath.
Some people complain of getting gassy from eating cucumbers and I've been told that the smaller ones have smaller seeds which do not cause this reaction as much. If I pick a large cucumber I peel them, cut in quarters, and carve out the middle part containing the seeds.
From my limited experience, generally, if they're still green, they're still good to eat. However, the more mature they are, the thicker the skins may be and the more well-developed the seeds may be.
Some varieties seem to taste fine even after yellowing to ripeness, though (e.g. Monika), while some are extremely bitter. The soil may affect how fast and if they get bitter, too. It's possible that still-green cucumbers may get bitter if they are mature enough, but I think the variety and soil conditions have a lot of influence here.
To illustrate this, I had a friend in my county who grew awesomely huge, green cucumbers that were likely of the Bushy variety. She gave me one, and it tasted excellent. I saved the seeds and planted them the next year. They only produced small cucumbers, and if I left them on the vine a long time to yellow, they were extremely bitter. (I attribute the difference in size and flavor mostly to the soil.) The Jolly Green F1 cucumbers I grew along side them were not so bitter.
In 2016, I grew Monika and Beit Alpha (among others), and neither of them had bitterness issues. Beit Alpha never turned yellow, though, but Monika eventually did, and still tasted great. Both Beit Alpha and Monika outperformed Bushy and Jolly Green F1 considerably, too, but I did water them differently. EDIT: While Beit Alpha never got bitter, I should note that really old fruits I tasted later in 2016 tasted like lemon cleaner smells (but they weren't bitter, and the lemon taste made them great for new culinary purposes). I think the lemon taste is the plant's reaction to anthracnose trying to make the fruit rot; the lemon-tasting stuff may be in order to fight the pathogen. That's just a hypothesis, though. But, there was anthracnose trying to infect at at least some of them (but it never went deeper than the skin in 2016).
Leaving fruit on vine longer often does mean the plant will produce less fruit than if you pick them sooner, with many, if not most, plants.
It is a matter of choice and experience. Try both options and see what works best for you. Personally, I prefer smaller cucumbers for the taste. The advantage of letting it grow is you get more vegetable but imo the taste is not as good.
