Former Loan officer-
I don't recommend a bridge loan, it sounds like you're maxed out on your debt to income ration if you have to close prior to buying, so a bridge loan will not likely work for you.
I would recommend a rent back. Continue to live in your home as you sell it and look for another, work with a good real estate agent who specifically understands your situation and can make a potential buyer and potential seller aware of it. Most people are very willing to work around this, as it's the case more often than not.
-remain in your home til it's sold
-set up your sales contract for your current home for occupancy taking place no sooner than 60 days prior to sale, if buyer doesn't agree, do rent back during that time
-set up your purchase of the other home to close on or as close to same day as your sale of current (this is where an agent working both ends helps)
-get everything in order as far as insurances, documentation, etc and do all the transaction with the same bank if possible to make it seamless
The most important thing is to protect yourself through the process and have everything in writing and legal.
It's not likely you're going to get around the situation with limitations on income and debt more strict than they have been a quite a while, but there's no reason you can't set up things with the old, shop for the new and make the change as easy as possible.
Best wishes!