This is something of an aside, but this is pretty important. Never run low voltage cables in parallel to high voltage cables for any length. Running a bare high voltage cable in the same cable tray as a network cable is a big no-no. To run them in parallel, one or both need to be in metallic conduit, not withstanding that high voltage lines should always be in a metallic conduit in any sort of commercial setting. Outside, overhead lines, not withstanding.
Depends what you mean by "high voltage." If his setup is strung like I envision it from the descriptions, then it's most likely insulated secondary feeds (600V class, actual voltage 120/240) from the transformer to both the house and the outbuilding. Obviously it would be very bad to try and hang the Ethernet from the secondary cables even if he does own them, but a separate attachment with physical separation, a grounded messenger wire, and an XLPE-jacketed outdoor cable would be technically feasible (can't speak to local codes, though).
Mutual inductance between the two cables will impart a low power voltage of equal amplitude to the HV cable. The two cables are effectively a 1:1 air core transformer. In most cases the power is so low as to not be a problem as the efficiency at 50/60Hz is very bad and twisted cables are designed for currents to cancel, etc. However, just because its probably ok, most of the time, doesn't mean there aren't plenty of ways for someone to make a mistake that is costly.
Last time I was involved in a data center, our design rules forbid HV wiring from even sharing the same cable tray, even when run in conduit.
Back to OPs problem. This is a matter of working your way down the chain and ruling out each. Pardon me if you've done any or all of these.
This assumes that when you say you have connectivity outside you out building, that you laptop can connect with it's inbuilt antenna and that you can get to the internet. Speed doesn't matter, so long as the connection is there. So, assuming this.
First, take your smart repeater into the house, with its regular, stock antenna and make sure it is connecting to your network. I assume there is some way to get to the management interface of the repeater when connected to it via a wired connection. If this doesn't work, then the problem is something in the repeater.
Connect your laptop to the repeater wirelessly and verify internet connectivity. At this point, you will have confirmed the configuration of each link the in chain under "optimal" circumstances. This means that the problem isn't one of configuration of one of the devices.
Take the repeater to your out building, outside, with the stock antenna. Since your laptop can connect to your AP from there, and you verified that the repeater can talk to your AP in the house, the repeater should connect to your AP outside your building. Verify this. If it does not, then the repeater sucks.
It should be able to connect anywhere a laptop can connect using its internal antenna.
Assuming the repeater can connect with the stock antenna outside, connect it to the 8dBi antenna, still with the repeater outside the building. Does it still connect to the AP? Can you still connect to the repeater with your laptop?
Assuming everything works to this point, you've proven the configuration is good, and all the devices are good, and the wireless link is good. Now take the repeater inside the outbuilding. I expect that the ability to connect your laptop to the repeater will fail here. Looking at it, it is a dual antenna unit, but what that generally means is that the unit will select the antenna with the best signal, but only uses one antenna at a time. Meaning that if you connect it to the external antenna, all its signal will be outside an all metal build and none inside. I am assuming this is why you were trying to use the SR600EX outside, with a ethernet cable to the SR300 inside. This would be the "correct" way, but does add another device into the mix. If you plug your laptop into the SR600EX with it outside the building, do you get network connectivity?
TLDR: Make sure all the devices work when they are sitting in the same room with all the cables connected -- working your way down the chain. Then start moving things to the outbuilding.