I'm in NE Ohio, and don't keep the house very warm. My seedlings, however, are in a room with a space heater. They're still pretty small, so no buds yet. 'Nuff said about THEM.
My overwinters, however, stay in the front, south-facing window and thus get full sun when it's out. Periodic problem during the winter was full sun when there was snow on the ground - the direct sun through the window plus reflection from the ground was actually too much, and I had to move them out of the window for partial days. Until recently, they would bud and then the buds would fall off. Then for a couple weeks, the buds would stay on and bloom, but then the flowers would fall off. However, in the last week or so the blooms are now sticking - and I now have pods starting. So they CAN pollinate and develop pods when kept indoors, you just have to make sure you move then a bit, either by blowing a fan on them periodically, or by gently shaking the stems. I think the more consistently warmer weather is part of what made the difference for these, as earlier in the winter the temps in the window varied greatly depending upon whether it was overcast or sunny.
I would imagine that if you started your seeds early enough, they would also get to a point to start pods indoors.