loosen the motherboard screws. they might be slightly out of alignment.

Cheapo stuff was and is always good for surprises.
Don't worry about that little bit of flexing, as there are no flexing-sensitive BGA.

Well sometimes bent motherboards can be bad news. It might cause damage to some crucial features, such as chips. I guess that's why motherboard manuals tell you not to screw it in too tight.

I have one of these (VIP CGA-III/P) and can confirm the OP's problem. I tried it in two different AT systems, and it won't fit inside either case. I had to remove one of the motherboards entirely from the case in order to get the card in so I could test it. (Working OK with an IBM 5163 monitor.)

Welcome to the wild-west of loosely standard vintage PC hardware. I have motherboards the ISA slots of which, regardless of what case I mount them in, do not line up perfectly with the expansion ports. Shit happens. Don't brute force it in, but if you are gentle but persistent enough it will ease in, however awkward it might get. Kind of like sex.

If you inspect the ISA slot closely, you'll see that it is actually not soldered correctly. It's not an issue with the card. On the left, the distance between the slot connector and the motherboard is wider than towards the middle. On the right, it gets wider again. I've encountered this with many mainboards. Often, you fully insert a card but the top of the bracket is still floating in the air, or as with your case, vice-versa.

You can try remove all screws from the bracket and leave only one at the top not fully screwed: then insert the card in the slot , make sure it sits OK and then tighten that lose screw on the bracket. Chances are it will fit OK with the extra wiggle the bracket will now have...

The bracket is soldered, not screwed, to the PCB. There's no easy way of removing or loosening it.

The ports are soldered - the bracket is screwed on the ports 😀

I have filed off a little bit of the rearmost edge that goes in the slot to reduce the stress. But I first just loosen motherboard screws while cards are in, wiggle stuff into the best possible alighnment and tighten the screws again.

AT motherboard/case fitment is just a hot mess generally

especially in the critical dimensions of "slot to board edge" and "board edge to case rear"

Right, I got the bracket off, and comparing this card to another similar-sized one, the problem is immediately apparent: the PCB is too long

It's only a difference of 2 or 3 mm but that's enough to cause a problem.