You can make the structure pressurized to match water pressure.
Then you do not need to worry too much about structural integrity, and make it as large as you want.
The cost is multi-hour (de)pressurization required while entering or leaving the facility. This could be a useful security feature as well: to prevent unauthorized access, and unauthorized departures.
The movie DeepStar Six (1989) has an example of explosive decompression.
Here is an RL example of the setup necessary for an unauthorized diver to reach your deepwater facility.
You can build it in pieces, but single large and round structure will have better volume/wall area ratio.
It will cost a lot, so research should be extremely high-value, and do not plan any large open areas in there. It will be a submarine.
Edit: I did mean to say that underwater structure has air pressure comparable to water pressure outside of it. I believe this can be done as part of Saturation Diving:
the divers live in a pressurized environment, ... [e.g.] an ambient pressure underwater habitat.
I still feel this is overkill for the goals. A better way to hide a secret lab underwater is to get a nuclear-missile submarine, and replace missile silos with labs and extra life support. It cannot go deep, but it can move around, it is designed to stay stealthy, and it can stay underwater for months.