The esteemed Jeff Cooper has praised the humble side-by-side shotgun ("coach gun") with exposed hammers for this role. They are relatively small for a shotgun, are not particularly expensive on the lower end, and are reliable to a fault. They can also be stored indefinitely, loaded, with no spring tension. Placing it into operation would merely require cocking the hammers. I don't agree with Cooper on everything but I do agree with him on this, and I own such an item myself for that reason.
Another shotgun consideration is that at conversational distances, there is little difference in terminal effect between a 12 gauge and a 20 gauge. There will be a considerable difference in perceived recoil, though.
If you opt for a pump (and this is by no means a bad choice), practice with it. An unexpected visitor in the night can be stressful and fingers grow clumsy with fatigue and excitement. I've seen otherwise skilled people drop shotshells or load them backwards even in relative calm conditions; dropping ammunition is inconveniant, but a backwards shotshell in the magazine tube will take the weapon out of action. "Brass to pinky" should be the mantra and should be practiced until committed to muscle memory.
00 buck and larger are not necessary indoors. I've been hit with birdshot and no longer consider it adequate. Some consider #4 to be optimal. Stay away from specialty rounds and steel shot.
There is some concern about magazine springs going bad if left compressed for long periods of time, even though the spring manufacturers tell us that this doesn't happen. My own experience has shown me that the springs don't mind long compressions, but the magazines themselves (whether boxed or tubular) have a tendency to accumulate dust, dirt, congealed oil and corrosion over time if left loaded which can impair function. Whatever weapon you opt for, consider weekly inspections and monthly shoots to be a bare minimum -- and more is definitely better.
Being skilled with a poor weapon is preferable to being unskilled with an excellent weapon. Whatever you choose, find instruction and practice. Practice should include reloading in a hurry and clearing malfunctions, not just shooting.