Unforunately, the 'Rec' button is part of a scanned matrix which makes it tricky to work with, however, now I can see the schematic there is a potential solution using the existing processor. What you need to do it mimic the 'Rec' button being pressed before the 10 minutes has elapsed. Before going into detail, the way it works is when the button press is detected it does something in software to 'top up' the timer for a further 10 minutes, it doesn't directly change any of the electronics, it just alerts the processor to take appropriate action.
The RST pin is the master reset input, it basically sets the IC back to a known state when the power turns on, it can't and shouldn't be used at any other time.
The INT pin is the interrupt input, its a hardware signal that diverts the program flow to a special routine then continues from where it was when that routine finishes. It isn't possible to see where it comes form on that section of the schematic but it is probably a sensor somewhere on the device. You can't re-purpose it anyway as the software it triggers will be specific to handling what ever caused it.
Back to the 'Rec' button, if you had an oscilloscope to see the signals it would be fairly easy to fake the switch closing at fixed intervals. Without knowing the signal polarities or timing, it would be safest to do it with a small relay and timer circuit. What you want to happen is the switch closes for maybe 0.5 second after say 9 minutes so it prolongs the timeout period. To do that you need two timers, one for the 9 minutes and other triggered by it to give the 0.5 seconds relay time. It can be done in just a few components and for little cost. How confident are you at opening up the device and performing some invasive surgery? I have no idea of the component layout so I can't say exactly where you need to attach wires but I can tell you where the points are on the schematic if you can trace them yourself.
Brian.