OK I misunderstood. Are they both repeating the main router signal or are they daisy chained?
I'm assuming the one getting 75 meg is using 2.4 ghz with 40mhz channel, in which case it is getting 100% of repeater mode throughput. The one inside the shop may not be able to connect at 40mhz so is using 20, and 35 meg is pretty much max for repeater mode on that. Or it may be the one outside the shop has a 5ghz signal, just a weaker one, the one inside the shop almost certainly is using 2.4.
Ethernet is going to give you the best and most consistent performance (should be able to hit your full 300M internet with 5ghz devices in the shop) but a wireless bridge should get you similar performance at that distance too. Depending on the quality of the devices you get a pair will range anywhere from about $50 to $120. The ubiquiti units are kind of the "gold standard" but there are cheaper ones out there that should work fine at that distance too.
If you do bury ethernet you need to run it in conduit or get direct burial rated stuff. You might get away with burying standard outdoor rated wire, the longevity of it may be impacted though as water will eventually infiltrate it.
You could attempt powerline adapters too, those can be hit or miss, depending on the wiring.
Is there a window in the shop you can put the router near to get better signal?