I can't really help with the what to do in the gap thing, I was relatively unusual in that I finished my master's in the July and started my PhD under the AHRC NWC-DTP in the following September, and so was applying for AHRC funding around the Christmas period during my master's. They took for granted that'd I'd complete the master's, but I had a BA with a 3rd year and graduating average in the 80s, and so had a big boon in that sense.
My AHRC research proposal seemed at the time like the hardest thing I'd ever written but it took me a little over a week. Things have changed slightly since 2014 (they fund fewer students now, and they only funded about 14% anyway!) but at the same time there is a tendency to chronically overthink this stuff. I couldn't, with the best will in the world, have done 30 drafts. I wrote it, left it a few days, re-read and edited it, and sent it to the funding nerds in my dept for them to glance over. A couple of minor changes and then it went it.
Make use of the help available, take your time. Think about what makes your research worth funding - what is the potential impact, how is it relevant? They really like things that ar en vogue or link to things that are en vogue. Be precise and spend a bit of time, but don't obsess over this stuff, honestly.
As above, you will definitely need the distinction to have a hope of funding. Best of luck - I hope you get it!