AWHome wrote

Do i need to use a router inside the barn instead of an access point?

No, you don't need a router inside the barn. You already had the correct idea how to set up such a network by using two CPEs and an AP in the barn.

The follwoing should work:

• Archer in your home defines the IP subnet, for example 192.168.1.0 (or whatver you did choose).
• CPE #1 connected to the Archer's LAN should be set to a fixed IP (e.g. 192.168.1.253) of this subnet – just to make it easy to reach its web UI, no other reason.
• CPE #2 at the barn should be set to a static IP, too, for the same reason. For example: 192.168.1.252. It must be different from the CPE #1 IP.
• Either one of the CPE must be an AP, the other one a client. Doesn't matter which is the AP.
• AP in the barn wired to CPE #2 should be set to a static IP, too. For example: 192.168.1.254.

Your PC/laptop/smartphone now receives its IP from the Archer by DHCP. You can test whether it works with a laptop either connected wirelessly to the barn's AP or wired to CPE #2.

If it doesn't work, try to ping the CPE #2, then CPE #1, then the Archer. Do you have IP connectivity? If not set your laptop to a static IP and try again. 

If there is IP connectivity, check the DHCP service:

Does it assign your laptop an IP? Does it assign the Archer as the default gateway? If not try to find out why DHCP is not working.