Thanks for the update Stefano. The 813 is a great tube for this I think, very robust and will handle what you have to give it. Many years ago I had a matched pair of RCA 833's with gold grids I used on a transmitter...that's what the HV power supply was for. Later I traded them for a Diamond Back bicycle for my son. Wish I still had them as the prices for them now are through the roof. The Gu81 is similar only a pentode. The 12au7 is one of the all time great audio tubes in my book...Think you will be fine with that.
Nice size tesla coil your friend built! Way fun!
Look forward to your next update. ~PJ