Fly Sparge Mash Out Question

One my 3 tier gravity fed system I have had problems with my mash out. I typically raise my sparge water to 175* and add it too my mash which is about 150 to 152 after an hour mash. But as I start the sparge my water in the mash tun (a keggle) drops to about 140* every time before it comes back up towards the end of the hour twenty minute sparge to get my 12 gallons. I can't heat the mash tun because I use a manifold and grain sits on the bottom of the keggle. It would burn if I turned it on to raise temps I imagine.

What are my other options to stop the mash and get my temps up to 165*? Should I add a few gallons of 175* water, stir and rest to get it up?

One option is to heat water for an infusion mash-out separately from the sparge water, i.e., hotter than sparge water, to get it the mash up to mash-out temps.

However, I wouldn't be too afraid of direct heating to mash out. With a burner on a low setting and constant stirring, you shouldn't have too much of a risk for scorching grain.