what would stop you using a buck converter ? like a LM2596 you could turn it to what ever wattage you need ??

Overkill

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1986 Mayfair (998 auto), 1974 S, 1974 Van, 1972 PMG Van

Too many minis

The overpriced buck converters from Jaycar are only $10, Xc4514.
They're not big, are easy to get, and are not any harder to wire up than just a chip.
You won't be able to jam one into the old regulator case, but it fits easily behind the gauges. You also need to protect it from contacting anything conductive, I used a small bit of conduit.

I have one on my Temp gauge, and slightly altered the voltage to get the normal operating temperature to be in the centre of the gauge.

I'm not an electrical engineer.. but I have played around with these before.

I can see there might be 2 issues with a buck regulator:
-produce a noisy DC output (works on average output voltage rather than a linear voltage)
-only support a relatively low current (<3amps)

Not sure if either of these are an issue for the circuit.. guessing probably not but worth calling out.

A possible alternative to using a LM7810. May help if you can't source 7810s.
Before I even knew of the existence of the LM7810, I had used 7805s for a few projects. In 2002, I used a 7805 and 2 resistors to boost the voltage output to 10V from a voltage stabilizer on my Triumph Spitfire which uses Lucas components in its electrical system. The stabilizer looks identical to my Mini ones. I followed the instructions from an old 1980 Dick Smith Electronic catalog, see the photo. Calculated the resistor values from the 2 simple formulae and used probably 1/2W resistors with values as close as possible to the calculated ones. Fitted the connected bits into the gutted stabilizer case, it's still functioning today.

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Clive G
1969 Mk II Cooper S
1970 MK II Cooper S
1963 Triumph Spitfire
1982 RX-7 Series 2