Regarding your tube mounting, from Radio Designers Handbook, Langford-Smith; "Where it is necessary to depart from vertical mounting, ............................................. the plane of the grid side rods (or major axis of the control grid) of indirectly heated valves having a high transconductance and/or a long unsupported cathode, should be vertical. This reduces the possibility of filament- and cathode-to-grid short circuits and microphony in filament valves."
6s4a data sheet; http://www.tubebooks.org/tubedata/HB-3/Receiving_Tubes_Part_2/6S4-A.PDF
I don't think you would class 6s4a as particularly high transconductance, so this is probably not an issue. Don't quote me, but I think the grid plane would be horizontal in your scheme, I'll let you decide that one! Maybe someone else has mounted them horizontally? As you point out, heat could be a problem, but until you try it, you won't know with that either!
Is there some reason that you can't spin the boards by 90 degrees and have the tubes side by side? After all, the boards are shorter that way.
The other consideration is the sideways force on the socket, but then these are short and light enough for that to not be an issue, and I don't suppose you'll be bouncing your chassis up and down! (Larger tubes sometimes have additional support.)