> Cranking current for cars is something like 100A+ IIRC
Yes, even 200A, I have seen 400A on specs.
2002 Accord MT starter should pull 90 Amps NO-load. The stall current will be far higher.
As a wild stab, a 15 horse engine might start with 1/10th the ooomph of a 150 horse, so 10A-40A. OTOH the starter fights each cylinder one at a time, a 150 horse 4-banger only has 40 horse per cylinder, so the peak current may be 1/2.5 or 40A-100A. On the third hand, cars must start in sub-zero weather, a grass-mower only far above freezing. Accepting JR's "100A" estimate for small warm car, probably not over 40A.
> marine grade momentary switch rated at 30 amps
Surely good for a lawnmower. Put 40A through a 30A-rated switch, it will work many times. The rating assumes possible re-start far out to sea with a squall coming; if the mower burns its switch by the back fence you walk to fetch the jumper-cables for a sparky-start.
> switch I've got on hand is 10A
Will work once or a dozen times. If you have a box-full, it might get you through for a while. Possible worst-case is that it welds itself, starter won't stop starting, battery drain and wear on the pinion (very worst-case: motor rotor bursts from over-speed).
Permanent(?) solution: auto-parts, ask for "starter solenoid for a 1966 Ford Mustang 289CID". Ford used the same ~~200A relay from the 1940s into the 1980s. Because it is SO common, it is not expensive and often in-stock. Ground (B-) the frame. Battery+ to "Batt". Other big lug to starter. Now apply +12V to the small terminal nearer "Batt", it THUNKs and the starter should whirr. Drawback: this oversize relay pulls about 1A by itself, not insignificant added load on the pint-size battery.
The key feature is two 1/4" -copper- bolts and a 2"x0.1" copper disk. The disk shorts the bolts. Disk, so that each new start tends to be on a different part of the edge to distribute the damage over 6 inches rather than a 1/4" spot. Two contacts avoids a fat flex-lead and gives some redundancy against welding (one side may weld yet the other side may break). A mower-hack can be less robust.