As you have oil pressure of sorts shows that the oil pump does not need priming.
That said you can prime the oil pump to be absolutely certain.
To prime the pump I would take out the banjo bolt of the oil transfer pipe (between block and filter head) and fill the pump through the banjo bolt hole. If you can turn the engine backwards whilst putting oil down the hole so much the better. Once you have put sufficient oil down the banjo bolt hole refit the banjo bolt not forgetting the copper washers and then spin the engine over with the spark plugs removed and see what the oil pressure is. All you can do is try.
If you feel that your skills in stripping and rebuilding your 1098cc 'A' Series then I think you are right to consider a ready to fit reconditioned unit.
I would try ESM first.
You may feel competent enough to purchase a reconditioned short motor (block. pistons, conrods, camshaft etc) and rebuild the motor from there.
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https://www.morrisminorspares.com/engin ... it-p830266
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Rebuilding a short motor is not difficult and can be done over a weekend. Just make sure that you have the workshop manual, gaskets and other parts required to hand before rebuilding. When you strip the engine you will soon see what new parts are required to rebuild the engine.
If you do rebuild a short motor you will look back and ask yourself what all the fuss was about. Do not forget we are here to help with any problems you come across.
Phil