I'm in the BIABB camp. I like having the basket to better contain the bag and don't have to rely on clipping the bag around the outside rim of the BK. It makes it easier for me to hoist out the grains with a rope & pulley.
I had the same thing happen, with a basket that protruded about an inch above the rim of my BK. During mashing I would set the lid on top and wrap aluminum foil around the kettle to block the gap, but I was still losing 3 or 4 degrees during a 60-minute rest, despite using a blanket to insulate the whole thing. The reason I had this problem in the first place was that the basket was an afterthought, and not designed to go with my shorter kettle.
So I cut 1-1/4" off the top of the basket and replaced the handle. I removed burrs so the bag won't get snagged. Now I can place the lid on my BK with the basket inside. Granted, cutting off the top of the basket removed its rim, so it is now a little less rigid, but that shouldn't be a problem.
I also made an insulating wrap for the kettle, using several layers of Reflectix insulation. I haven't tried the insulation yet, but hopefully it, plus the removal of the gap, will keep my mash temps more consistent.