Basically the transistor is being used as a voltage regulator, so that the output is regulated roughly to whatever is on the setting of the variable pot. I actually have some of those transistors around still, NOS, doing nothing.
You will need a few changes to get to 300VDC rating. First would be a small isolating transformer, or easier is to get another 38V secondary transformer of around 10VA rating and use it in reverse to make an isolated 115VAC supply at low current using the existing transformer as a 38VAC source. The diode bridge would have to be a 1000V 1A unit, and the capacitors would be around 47uF 400V each. The pot would have to be a 20k to 100k 5-10W wirewound or carbon film pot, just to get the voltage withstand ability. Digikey CT2160-ND is a likely candidate. Caution it will run warm. Knob and such is your choice.
Bear in mind at this voltage contact is very likely to be lethal, so be careful. No second warnings, you should not be doing this if you are not at least somewhat experienced. Plenty enough to make you very dead at this voltage, at best incredibly painful.
You would basically exchange the meter and all the associated resistors and shunt switches to the negative rail, as you will have to use a high voltage high power transistor to provide the power supply. High voltage PNP transistors are both very rare and very expensive.
You probably will be using a NPN transistor instead, likely a BUX85GOS-ND ( digikey part no) in a TO220 package, isolated from a small heat sink, as the pass device. Then it will be upgraded to a 300V unit. Power rating will be limited, and you would be best to place a 4k7 0.25W resistor in the base lead to provide some current limit ability to the supply. In any case it will not be a precision supply, but will be reasonably stable. It will not have a very high current capacity, as there only needs to be a limited current ( mostly from the transformers impedance) to initially charge the capacitor under test.