Your schematic is upside down
Please draw high voltage up, ground down in the future.
I think the only advantage to be gained from using such device is elimination of sparking and reduced wear on the mechanical switch, are those things a problem for that guy? Perhaps a slightly stronger spark by elimination of the snubbing capacitor, but I'm not sure.
As for your design, it applies (almost) full coil voltage to the switch through the BE junction of the tranny, albeit somewhat current limited by the resistor. Not sure how it will behave sparking-wise with a real coil instead of an LED and whether you gain anything.
As it stands, it frankly isn't going to work at all, because it needs to get to a point where several amps flow though the tranny and its output voltage is barely above zero. Go calculate how much base current is needed for that and whether a 10k resistor will do.
No, really, switch it to an NPN and add another inverter stage in front of it. Its output will reach closer to zero, less heat will be generated and base current will be independent of coil and ballast resistor voltage drop.
And actually, N MOSFET might be better still. Even less voltage drop if you get a good one, no heatsink needed, doesn't need overvoltage protection if you get an avalanche rated one.
I've actually built a spark generator using automotive ignition coil once and used the IRFP4768 to drive it. A bit overkill in retrospect, a cheaper part will probably suffice with slightly more heat generation.
Power dissipation can be calculated, just assume that the tranny is conducting full coil current continously and do the math with RDS(on).
A few nF capacitor may be needed from source to drain to avoid exceeding dV/dt rating of the transistor when it turns off. I calculated that 1nF sufficed for my use, IIRC.
I've heard mechanical contacts can oxidize when switched without load. So for sensing if the switch is closed or open, pass some generous current through it. Maybe 10mA at least? I'm not sure, perhaps someone will know.