I've got a couple of these timers I've acquired over the years. First the plastic box that houses the timer can't weigh over 75 grams. The 3 position switch is a very nice switch, takes a bit of torque to turn. One of the timers I have has Velcro on the back, I've never tried that. The power is a 9V power supply. The power, the probe, and the footswitch all come out of the "bottom" of the timer.
The footswitch is what resets the timer to zero. As long as the timer is plugged into the power supply it's counting. The footswitch is a beautiful Treadlite switch, smooth. If I were to permanently set up the timer, I would as Matt suggests use the footswitch as a big button or bump it with my hip to start.
For you you might be able to use a switch for the little power supply. I'm not familiar with the inner workings, but I think the footswitch just briefly interrupts the low voltage. The probe and the footswitch use telephone jack connections. If my eyes didn't deceive me, on the footswitch, it looks like only two conductors are used, 1 black wire and 1 red. I think the footswitch opens up the low voltage circuit when depressed. The probe looks to have 3 conductors wired.
I would be darn sure that little plastic box didn't go ballistic into a tray. These little things work well, now if they would automatically replenish the developer