As you have already worked out, they consider the birdhouse to be their hive/home. They will not leave it willingly. If you put it inside a bigger box, they will just try to fill up the gaps with comb again but they will keep using the birdhouse. You cannot stop them from doing that, they are just being normal bees! 
I am worried that you will end up with a mess if you do that. I would suggest a cut-out instead. Take the comb from the inner cover and rubber band it into the empty frames, like this:
https://forum.honeyflow.com/uploads/short-url/wfqpuoZIfYezHCmZHtm5kzCS4vi.jpg
Not quite sure what you mean by “forcing them out”, but I would open up that birdhouse. Then I would try to cut out the wax comb inside it in as big pieces as possible and rubber band them into frames also, putting the brood comb in the middle frames and the honey and pollen stores on the outer frames. Yes it is somewhat destructive, but unless you can build some kind of adapter to mate the birdhouse with your Flow hive, the bees are not going to use your Flow hive properly for you.
Just my thoughts. Others may have better ideas. 