IPhone XR take a second to go to sleep, instead of going out instantly

This is, unfortunately, expected behavior. Whenever you activate an iPhone (which you do as part of an OS install) it talks to Apple's servers and does various things like exchange crypto certs for APNS, etc. It also checks the IMEI of the phone to see what carrier Apple sold it to, and validates the SIM is valid for that carrier (if you buy an unlocked iPhone in a country where you can buy either locked or unlocked phones the vendor will actually copy down the IMEI number of a normal locked phone, and then enter in their system where it gets pushed to Apple's servers as an unlocked phone, there is no physical or configuration difference on the phones).

The easiest way to get it activated is (as you surmised) to use an AT&T SIM. You mentioned you have an iPhone 4, you can activate it with that SIM. There are plenty of companies that sell microSIM to SIM adapters, but if you want to do it right now you can just place the micro SIM directly in the iPhones 3GS's SIM try with tape. I would not recommend doing this on a device you intend to keep the SIM in for a an extended period because the SIM is likely to come loose (and jam the SIM tray), but if you are just inserting it, activating it, and removing it you should be fine so long as you are pretty careful with how you apply the tape and not shaking the phone too much.

Although it's likely a hardware issue, you could try a few things if you don't want to get hardware help from Apple yet.

1. See if this happens when booting in Safe Mode.
2. Install a clean OS on an external drive to be sure it's not some extension or conflict.
3. Look in the console logs - see if you can see any obvious errors.
4. Try with other monitors and cables - it could be a bad cable or display. An EFI or firmware update might be needed to make the mac handle this situation better (it should never freeze) - and a report to Apple with details how to reproduce this might help get a fix sooner unless others have reported this specific combination.
5. If you have alternate RAM - do try swapping that in for testing. The newest macs use RAM heavily for display buffering - and wake from sleep is a time when you first can see issues with RAM going bad.

I'm lucky to have stores close by and would head there for a diagnostic checkup - so keep that in mind - it could just be a hardware failure / bad RAM, but anything you can do to narrow down the cause or rule out software / accessories by disconnecting everything but the bare minimum could save you a trip to the shop.