Yes, the BeePre will have a severe problem with non-standard current draw. At 1.5 amps, the 300B will see only 4.4 volts, well below what is required for the full 60mA current draw. I doubt Jac would honor a warranty under such conditions, and he would be completely correct in my opinion.
Of course the circuit can be modified, but it would require more current from the filament power supply, causing it to run hotter and thus be in danger of overheating.
The BeePre is not designed to accommodate non-standard "300B" types which do not meet the specification for 300Bs.
The only non-standard tube that will work correctly is the 6A3, or a 6B4 with a different socket. That is not so much by design as a fortuitous accident!