Can you ask the indoor cage staff if you can place a damper over the light? I think I know the lights you are talking about. The place I go to got new machines a few years ago - choose your velocity, multi "tire" types (vs. arm-loaded), with a big ol' red LED just to the left of where the balls are released. I talked to one of the guys working there and he mentioned the same thing (he's a full time lefty hitter, and the light is to the left of the ball release, so he's looking at it all the time). What he said was that there is a little darkening filter in there that sometimes falls down/slides down so you are seeing the unfiltered LEDs, which are obviously very bright.
Do you have a garage or a basement large enough for you to take swings? You can get a 7x7 net like this one off Amazon (I have it). I'd be comfortable taking swings off a tee using this in a basement or garage, where I wouldn't care much if the ball somehow missed the screen (if never has, with the tee set up about 2 feet in front of the screen). I'm mostly concerned about the few times the ball misses the net pocket and bounces/rolls back behind me.
Once I have a garage (next year), I'll be using it indoors for both short range throwing drills and some indoor tee work. And it's portable. You just need a couple of sandbags to hold it down.
This is the one I just purchased FWIW https://www.amazon.com/gp/product/B06Y22YLQ2/ref=ppx\_yo\_dt\_b\_asin\_title\_o00\_s01?ie=UTF8&psc=1
I think it's this one: https://www.amazon.com/dp/B06Y22YLQ2/ref=cm_sw_r_cp_apa_fabc_wiHRFbXGYBTG1
I always recommend this to people. https://smile.amazon.com/gp/product/B06Y22YLQ2/
For the price, you get a net you can drive on and never have to chase discs. If you don't want to buy a basket, you can actually just use the included baseball strike zone marker and position it at the height you want to practice putting. It is more than enough to work on practicing your putting aim. Just pick a square and aim at it. Don't worry about "catching a putt" since if you can consistently hit a square from 15, 20, 25, 30 feet, you'll be able to apply that for hitting chains.
I don't have a basket, I bought this instead. https://www.amazon.com/gp/product/B06Y22YLQ2/ So I just grab my stack of 10 and aim for the strike zone. Throw all 10 startting at 15', then 20, 25 and 30. Do that a few times a day while working from home to loosen up and stretch the legs a bit.
Can also use this to practice my driving form in my basement. I stand 20 feet from the net and use putters to drive. No issues with it not catching. Probably my best disc golf related training purchase. I do have a basket ordered finally, but for the price, this thing has been such a useful tool.