Have you considered using a "Shaved Door Handle" kit?
"Door handles are off - but you know I'll never miss 'em.
They open when I want with a solenoid system... "
Apologies to the Beach Boys

Battery could become an issue, but with an external Jump Plug easily overcome.

With a touch of a button on key fob (or placed button) door opens.
Solenoid could be linked to the sliding rod mechanism you mentioned.

Cheers - Jim