You could try something simple like this - at least to get by.
Use an N-FET with a slow transit of the linear region to control the current during startup. Essentially the same thing as the Linear part LTC4397 (which I have used successfully), just much dumber and without all the protections.
Green trace is the gate voltage.
Blue is the load current. The load is R3 set at 10ohms as a wild guess.

The voltage divider would be set for the appropriate gate voltage, the cap C1 sets the rise time of the gate which will determine the rise time of the current. It's simple and dirty.
LT Spice attached