Thermaltake TR2-430 has a rated wattage of 430W, so you could run all four 6336 tubes when they are heated as long as you run two of the 6336 off of 12V rail 1 and two of the 6336 off of 12V rail 2. But you're right about the inrush on power up; some switchmode power supplies will fully shut off when they go into overload, and only turn back on when power is manually cycled off then on again. Other supplies will keep trying to push their max current, and just turn on/off a whole bunch over and over (producing a huge amount of high frequency noise) until the current draw drops below their max rating. If the TR2-430 is of the 2nd type, then it could potentially start with the high inrush, and the inrush will decrease as the tube filaments reach their operating temperature. Modern computer power supplies like that are protected against faults like current overloads, so I don't see any harm in trying to see if you can get it to start with the 4x 6336.
P.s. 4x 6336 - Able to turn your furnace off during the winter
lol!