Hi Darryl
Your genetic genealogy question has a more straightforward, neater, resolution than the one that you propose. So I'll recommend the approach you take and only then comment on the 12.5%.
If your deductions are correct, both you, your putative great grandfather and his second living relation will have essentially identical Y chromosomes. The two of you can have Y-DNA tests and if you are both all-male line descendants of the same great grandfather, then the markers will be identical -or at a very outside chance, just one difference in just one of the markers. You could both order the basic 12-marker test but it is not offered any more by most companies. Most offer a test with about 40 markers and this would be good value -expect to pay just short £100 each. Companies also offer tests with more markers but for your purposes, I think this would not be needed. You can always upgrade and get other genetic genealogy tests done from the same sample if you become interested in the subject.
Different companies report Y-DNA results is slightly different ways. The results from one company can be converted to those for another but I'd recommend you both have the test done by the same company to simply matters.
There are many companies that offer Y-DNA tests. By far and away the biggest is Family Tree DNA but there is also dna.ancestry (used to be called Relative Genetics), DNA Heritage, EthnoAncestry, familybuilder, Genebase, GeneTree (I used them because they offered the best deal at the time), Oxford Ancestors, Pathway Genomics, Personal Genome Project and Trugenetics. There are more. Check them out on the internet and compare costs and the extra features that some companies offer. Check for surname projects - Family Tree DNA will offer a discount if you have your tests done under such a project. You may get a discount and/or extra features for having a pair of tests done together.
You are right that if you don't share any other recent ancestor, then you will share about an eighth of your autosomal (non Y-, non X-chromosome and not mitochondrial) DNA. It would be less helpful and more expensive to test your autosomal DNA although the results would probably be more exciting! If you want excitement look at the tests offered by 23andMe and deCodeME.