Go ahead and try it.
Well, I do have experience with using chamber adapters – I really like them, and I do have a Spanish 1916 Mauser chambered in 7.62x51 that I also considered converting to 7.62x39 – BUT I passed because 7.62x39 wouldn't feed from the magazine correctly. For that rifle I load my own x51mm that are less powerful – and since the Mauser barrel slugs out at .315 – I use large Mosin-Nagant sized cast bullets.
If you don't mind loading each cartridge one by one in the Mauser – go ahead and try the adapter. They are super cheap, and your Mauser will easily handle the extra couple thousandths in bullet diameter. Have you slugged the barrel yet? If it slugs large like mine, it might be much more accurate with the larger bullets.
Adapters that are properly sized and seated, stay in place, and can save you a ton of money.
My grandson and I were shooting two rifles today with adapters. He was shooting a Czech VZ52, originally chambered in 7.62x45 and converted with adapter to 7.62x39. That rifle has had an adapter now for at least 10 years and 5K rds. Because of the low recoil, ability to rapid fire with significant power, it has been a favorite of my grandkids for years.
I was shooting an 1895 Brescia, originally chambered in 6.5 Carcano and converted with adapter to 6.5x39mm. The Brescia slugs tight – so std 6.5 mm (0.264) bullets suit it, and given the old girl is well over 100 years old, 1.5 cc of powder is more than enough stress on the old iron – and using the smaller cases saves me money on cases, and saves me the hassle of topping each load with filler.
I've also converted my Garand to 7.62x51. Worked fine. I shot it that way for about a year before converting it back to 30-06. 30-06 just seems 'right' for a Garand for me.