What are the constraints when choosing R value for a RC time constant?
Yes, basically it is always a tradeoff:

If the RC filter drives some high-impedance device (Opamp, ADC, NE555, ...)


If you set R too low, the circuit will consume more power, and the capacitor will need to be bigger, which can become physically impractical / too expensive.
If you set R too high, the flowing current will be too small and the small current flowing to the device "reading" the capacitor voltage (Opamp, ADC, NE555, whatever, ...) will become significant. Then your time constant will not be accurate anymore.


So choose the R value so it sets a current through the capacitor that largely (10x, 100x depending on the accuracy) dominates the input leakage of the "reading" device.

If the RC filter directly drives the whole load (your case, it seems)


If you set R too high, there will be a high voltage drop between the supply and the circuit, so the circuit won't work.
If you set R too low, the capacitor will need to be bigger, which can become physically impractical / too expensive.


So compute the higher R limit, which is (Vsupply - Vload) / Iloadmax, with


Vsupply: your supply voltage
Vload: the voltage required by the circuit behind the RC filter
Iloadmax: the maximum current that the circuit can draw


and the lower R limit will depend on your time constant and the maximum capacitor value you can afford to use.

However, you must then account for the load current itself when calculating the required capacitor value. Because the time to reach the target voltage will be highly dependant on the load current, not just the RC time constant. But honestly, I am unable to give quickly an equation for this. I would suggest making a quick simulation with LTspice, here. Or try the real circuit with several capacitor values, once you choose your R to some value just below the higer limit calculated with the above equation.

Now, what is a better idea

The problem is that, if your load have big current requirements, the computed max R value will probably not lead to a realistic C value. So, what people usually do is to use a soft-start circuit using a MOSFET. This way, at steady state, the resistance between the supply and your load becomes negligible (so there is no voltage drop).

Here is an excellent suggestion for this: How to limit inrush current?