I suggest flooding your dome.
No, wait, hear me out.
Your subs, essentially, want to be able to get in and out of the dome. Moving them in and out of water is messy and inefficient.
Your dome, too, will have to pump out water occasionally from a ‘sump’ (I’d call it a bilge, but this isn’t a ship), so you must have pumps capable of moving water out of the dome. Trouble with these pumps is theyll have to do an awful lot of work to pump out into 1000 atm, which will make them slow.
If the base of your dome is flooded with water at sea level pressure then you give the pumps a bit of a buffer that can happily absorb a sudden influx of water.
Your subs can then simply sail from the outside into a water lock with an inner and outer door (preferably ones that open sideways, or if you’re not just opening the doors to move water around, outwards). Once the sub is in you can close the outer doors and use the water pressure inside the lock to force water into the flooded base of your dome. If you allow the water out through carefully chosen vents you can minimise currents against the sub. If you want an example of this try looking up canal locks.
Once the pressures (very quickly) equalise you can open the inner doors and let the sub in. Your sump pumps can then take their time with the extra water, and thanks to how incompressible water is the level of your sub marine submarine pen won’t even rise that much (say your water lock is 20m by 10m by 10m you’ll be adding just under 100l of water to a lake a mile wide; or less than 0.1 mm rise).
Operating in reverse is similar, but this time you let water in from the outside, taking care that either your sub is strong enough to deal with the watery jackhammer you’re applying (hydraulic shock is not the same as water pressure) or that you let the water in slowly. Keep the lock open for when you want to return.
Plus you get a lagoon under the sea, and that’s pretty cool.