@rehashed 

I am doing something very similar to you with 225-outdoors and have been very happy with the result.  One thing you need to be aware of is that when you mesh, all APs must be on the same 5Ghz channel, and when you have a linear string of APs, especially with elevation/foliage factors, you can run into a lot of crosstalk when the edge APs cannot 'hear' each other with significant performance issues.  I solved this by adding a second 225 at the house, since that in my case is where the controller/router are, and then broke my linear string into 2 smaller strings operating on different 5.8G channels.  I placed my APs on either sidewall of the building to use it as a natural RF shield.  I was sure to enable 'preferred' root AP for each child node in the mesh in the controller to prevent flapping or unexpected mesh configurations.  I also selected channels in my region that permitted the highest power/bandwidth (149+ in CA).  To further optimize the BW available to Meshed APs, I disabled user access to the outdoor APs at the house on 5.8 (they are only permitted on 2.4) to prevent user traffic from competing with backhaul traffic, and added an EAP615 indoors to provide Wifi6 and operate on both bands.

The above may not work so nicely in your setup as my assumption is that in your architecture, you are using a cloud controller, and the modem and AP on the lamppost constitute the router/controller in your setup, and that AP is thus the root AP (since TP-Link hasn't found a good way yet to allow you to use the ethernet port of child APs without them thinking they are root APs because the etherlink is up).  You still have the issue of house-barn chatter possibly talking over post-house.  The run from post-barn could be ~50m, which might be just too far to get your 300Mbps reliably direct from the post.

In an ideal world, one could add a second AP at the house, mark the upstream house APs as 'child' to prevent it becoming root in the controller (this feature does not exist at this time!), connect their ethernet ports back-back which has the effect of bridging traffic between them, and then provision the second house AP which will self-identify as a root node due to it's active ethernet port on a different 5.8 and then lock the barn AP to that one as it's preferred root.  Since that is fiction...

You could temporarily power off the house AP, and test the performance from the remaining two APs (ie POST-BARN) and see if that meets your needs.  If it does, then add a second EAP225-outdoor at the lampost, connect it to the LTE router and configure on a different frequency.  Pair post1-barn and post2-house APs in the controller.

You could install a high gain cellular antenna on the outside of the house and move the LTE Modem there, add a second 225 AP at the house, pair 'em up and off you go.

Anyways, let me know if any or none of those ideas work for you :)

  @rehashed 

Moving just the controller won't help, you need a way to isolate the lamp and barn radios from each other (ie different radio channel, which requires another root AP in the mesh).  

I think the most useful test would be powering down the house and seeing what kind of RF link quality you can establish directly from the lamp to the barn.

There is also a chance that your mesh has already configured itself with both the house and barn homing directly back to the lamp (root) AP.  If so, you might notice a performance improvement if you force the barn AP to mesh with the house, instead of the lamp.  In either case you should also experiment with at least temporarily removing 5.8 user access from your SSIDs on the child APs (house and barn).  This will allow the mesh links to operate at their peak, the irony is that the 2.4G link will likely max out in the 100Mbps range.

The thing to watch out for is a 5.8 user at the barn generates a packet, this chews up a timeslot at the barn AP and blocks the channel for the other two APs.  Then the barn AP retransmits that packet, again another timeslot consumed at the barn and blocked elsewhere.  The upstream AP receives that packet, and a short while later retransmits, chewing another time slot, and blocking the channel.  Packet now received by root AP and forwarded via ethernet port.  If that packet s repeated, and the user generates packets which consume 20% of the radio channel aka link capacity, it's actually consuming 20+20+20=60% of the channel capacity.  This is why you want to minimize radio hops in your mesh and keep chatty users off as well.  If that barn user had used 2.4G for access, then the effect on the network would only be 40% instead of 60%.  Now add in all the house traffic on 5.8, you can see the mess.