bat tube amps relating to heat displacement

i have a VK-5i pre that does not create heat concerns. the aes cary sixpac amps felt like a blow torch so i thought i needed to go to ss for an amp. wanted to match the pre with a bat amp but before i do leave with world of total tube sound (which i hate to do) enter the combo world of pre tube/amp ss i wondered two things: first, would this type of bat pairing work well? second, would the bat tube amp be any less warm that the cary?

thanks for any thoughts.
If you think the 6 packs were hot, try a V12R! I hated to let mine go, but I traded it on a new Cary 120S amp, and it only puts out about 1/5 the heat. It gets warm, but not hot. I have 96 hours break in time on it and it sounds fantastic! I hope you can afford to stay with tubes. If not good luck with SS. Maybe a used McIntosh 7270. I had one for 18 years and it worked fine with a Cary SLP98P preamp. Not sure about BAT. Good luck.
thank you. i had a mc ss integrated 6600 and liked to look at it but couldn't figure out what all the fuss was about how "great" mc was supposed to be. then i heard tubes and for once realized just how good music could sound.

strange that in this world of ours two things haven't been figured out: one, how to make tubes run cool and two, how to get rid of all the stinking wires!