I'd be a bit wary of "gut feelings" about the housing market. It might be a good time to buy in some areas, but I can say from research in my own area of late that there are some areas where it is decidedly not a good time to buy anymore, if anything I think we're headed toward a bit of a bubble again. Disappointing for me since I did what you're saying, opted to lower my 401(k) to save for a down payment, regretting it now.
Besides, you're looking to live in this house, not rent it, right? In which case the real investment profit to you is potentially turning some of your rent into capital accumulation. Maybe compare your current rent to the money you'd pay for a mortgage and see when the amount of interest you'll pay (i.e. the rent on your borrowed money) is lower than your current rent. Then add a safety margin for extra expenses associated with owning and make the calculation from there. I've attached a sheet to help play with this calculation some if you like, it's based on an MS Excel template with a few additions I made to compare more easily. Not professionalized so sorry if it's easy to get lost in, but maybe a good starting point.
I share because I did this calculation myself recently and, when I found a way to move to lower my rent some, came to the conclusion that it was in fact a very bad financial decision to buy now. The answer you come to in this is VERY location and situation specific, so don't take general answers and assume they work for you.
And on the bridge loan, I'm with willn, if you're in such a rush that you have to borrow from that then maybe it's worth delaying a bit.