If he is your last remaining direct male line, go to FTDNA. They are the yDNA specialists, and as you say you are only going to get one shot given it sounds like you are particularly interested in the direct male Britton line.
Yes LivingDNA will give you a broad haplogroup, but no ability to join projects for either your surname or haplogroup project and to compare yDNA results with others.
LivingDNA is autosomal DNA with a few y/mtDNA SNPs thrown in and is more focussed on ethnicity breakdowns than on matching, which it has only recently started offering, after many years of promises.
FTDNA has many many projects whose volunteer project admins (no affiliation to FTDNA, just DNA addicts) are a wealth of knowledge on either the surname or haplogroups or areas their projects are for, and generally advise people on what tests are most suited to what you are trying to gain from DNA testing.
FTDNA has been offering yDNA testing for many many years being the pioneer for Direct to Consumer DNA testing.
FTDNA also keeps your sample on file and it can be used for upgrades, sufficient viable DNA remaining of course, as you need different tests without needing a re-scrape.
If your candidate is elderly, I would suggest that you also contact FTDNA before ordering to ensure they include an extra swab and vial to make sure later upgrades may indeed be more likely, and start at the largest yDNA test you can afford of the y37/111 marker tests offered. If you get hooked and /or get fascinating yDNA matches, upgrade at a sale to BigY700 for the most precise placement on the yDNA tree of mankind, which placement keeps on giving as more people upgrade and the tree expands.
I would add FamilyFinder as well.
Although you cannot transfer that into Ancestry, you CAN transfer atDNA from FTDNA to MyHeritage and GEDmatch for free, with a small charge to unlock additional tools beyond basic match lists and ability to contact matches
Check Roberta Estes' blogposts on yDNA testing at http://dna-explained.com
there are many helpful posts there.
The uploads to multiple places you refer to are more focussed on autosomal DNA (what Ancestry, MyHeritage, 23andme, FTDNA's FamilyFinder and LivingDNA offer) but yes, there are places you can transfer your yDNA raw data results to for alternate analyses (yFull.com, The DNA Data Warehouse being two that spring to mind) and some others just for sharing results.
mtDNA isn't being held for ransom, it is simply not the most useful test for genealogical timeframe matching, UNLESS you have a particular problem to solve and can find candidates down from eg the two families you wish to compare on the direct maternal line. Only FTDNA offers mtrDNA testing, with matching, and again, projects.
Autosomal testing is more suited to confirming recent generations on your tree on all lines.
Hope this helps.
PS watch out for regular sales.