You might want to try a straight 30 weight. Tecumseh says that 10w30 is acceptable, but that HD 30 is preferred. My 12 HP Tecumseh manual also says that using multi-grade oils will increase oil consumption.
straight 30 is what is prescribed, yes, but while it is still available, it is getting harder to find unless its maji-mart brand or something. Yes, Wally world still stocks it, but 9 of 10 you have to dig for it, and I like the idea of a Diesel oil's detergency, and wide availability.
I just changed both of my mowers, a 10HP Briggs and a 13HP Kohler to 15W40 HDD Amsoil. I do not have an IR tester, sold it with my business, but the overall feel is good running and doesn't seem to get as hot as before even with higher ambient temps.
I'm using Bobzoil 15W40 in my Honda tractor right now. No issues at all. But then again, I didn't have any problems with what I ran before ... Mobil 1 10W30, various homemade blends, etc ...
If 10/30 is slightly consumed then 15w40 will likely be consumed also. There is a good chance that the oil gets hot enough to evaporate or burn off in small amounts. Tecumseh engines prefer a mono grade oil as the manufacturer states. I use a 30 wt oil in my 5 hp tecumseh motors that power my turf sprayer, aerator and verticutter. These engines get hot and work hard. No matter what weight i use the engines still use a bit of oil. I use 40 wt oil in the kids go kart with a 6 hp engine and yes it uses a little oil. I use my mowers on a daily basis and at around 20 hours on a new oil change, i still have to add a little makeup oil. If you really are that worried about oil use then use a straight 40wt and let me know how it works out.