Pretty sure an amp would be problematic. I don't know if a moca amp exists, but the signaling in moca is more complex than regular OTA type signals and an standard amp may cause more issues than help. Not saying it won't work, but it may be chasing your tail trying to find one that does.
The overall length of all cable on the home side of the node is an issue for moca. If you can trim length inside the home to the hopper and any other joeys it could help.
Is the remote building "router" connection to the main house wired or wireless? I'm not sure what you mean by "3 routers daisy chained." If they are doing some sort of wireless relay, then that will be problematic.
If Cat5+ wired, then
@TheKrell is right - put a switch in so the Joey does not go through the "remote" router(s?), or at least make sure the router is in access point mode and not routing. This should work, but 300ft is at the edge of 100mb cat5 spec.
If it is wireless, and rg59 is already in place then there are Ethernet over coax adapters that are not moca or deca and sole purpose is to push Ethernet over coax over long distances - over a mile in some instances. This might not be the cheapest but should easily carry a reliable 100mb or even 1gb Ethernet signal over the existing RG59.
A poor man's version the Ethernet over coax solution might work with 2 HICs directly connected, ie: homerouter--HIC--COAX--HIC--JoeyCat5port. This would create a single point to point connection and removes the node, hopper, and any home joeys from the mix.
If you do run new cable, then definitely go with RG11, not RG6. Incremental cost is minimal and should be more reliable.
EDIT:
You wouldn't necessarily need two HICs, you could do homerouter--HIC--COAX--JoeyCoaxPort. This would have the same effect of creating a direct connection with one less piece of hardware.
