Shortly after I enter the mechanist hideout, after Ada has opened the door, my game closes. Everything up to this has been running fine. My character is level 20, so I haven't been all over the map and it's the first of the DLCs I've started.

The answer just might be in the Fallout series general section, if you cared to search that place. Anyway...

Yeah, the Mechanist Lair is very poorly made. The recommendations for that area are three:

1. lower graphical settings, limit fps to 30 if you haven't already and remove mods that can be safely removed, then try to go through it. Might work, might not. Generally, if you have the unofficial fixes and avoid the 'scrap everything' type of mods, the crash should be fixed.

2. identify the area in which the crash occurs and pass through it without looking at it directly. I used this one. If you hug the right wall and strafe left along it without staring at anything else and go in the decontamination corridor, it should work. Just don't look back.

3. use the console to skip the whole thing and complete the quest or teleport to the mechanist directly.

This is one of those bugs theres no universal solution for. Even with community patches its a hit and miss affair as to whether this happens or not. I'd suggest loading up a save from before you enter though.
A wider search on the net will provide various tips and tricks to try but as I said, it appears to be one of those things. I just wish I could remember how I got beyond it the first time around. :/

Edit:honglath above me does provide some good advice though.

Simply port Fallout 4 to a new game engine, and then sue Bethesda for gross incompetence.

Yeah, that sucks, sorry there's no, "And here's a patch for this baffling abuse of a 20 year old game engine".

I ran into a different issue with Nuka Cola World myself (that may happen to some players but not others) where for some reason no doors could be opened anymore within the greeting maze. Had to clip through via console commands. *shrug* Bugthesda, indeed.

and then theres the "cascading lag" issue with the vault 86 (or whatever its name is) dlc which can carry over to others.

I just ran into this bug also. Tried all the usual tips to no avail.

As a 'last chance shot' I turned off all optimisations in NVidia's GeForce Experience - Magic!

I've just about completed the quest wth no further issues. And I've even turned everything in-game back up to ultra.

It may have just been a solution for me - but it might be worth a try if you have optimised your games using Geforce Experience....