Okay, here's my KISS design.
When power is initially applied, capacitor C3 turns on MOSFET M2, which pulls timing capacitor C2 to ground.
This keeps the 555 output high and applies full voltage to the solenoid.
The value of R2C3 give a delay of about 400ms before the PWM starts.
This can be varied by changing the value of R2 or C3.
The PWM frequency is about 1.3kHz with the indicated values of U2 and C2.
This can be varied by changing the value of C2.
Note that the solenoid current goes to its maximum at start and then settles to the hold current at about 500ms.
The hold current can be varyied by adjusting pot U2.
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