Setting up Windows Admin Center in a middle school

I am attempting to set up remote management of my campus computers via Windows Admin Center. I am starting with my main desktop workstation (referred to as Base) and my take-along laptop (referred to as Remote).

I ran winrm qc to enable BOTH Base and Remote to connect to themselves in Windows Admin Center but when they try to connect to each other they get the same error:

Connecting to remote server [Base/Remote] failed with the following error message : WinRM cannot complete the operation. Verify that the specified computer name is valid, that the computer is accessible over the network, and that a firewall exception for the WinRM service is enabled and allows access from this computer. By default, the WinRM firewall exception for public profiles limits access to remote computers within the same local subnet. For more information, see the about_Remote_Troubleshooting Help topic.

I can verify that the names are valid (I assume) because they can be found when searching AD in the WAC tools when adding a computer.

The outputs are basically the same when run from either Base or Remote.

To verify the computer is accessible over the network:

> ping Remote
Pinging Remote with 32 bytes of data:
Reply from Remote: bytes=32 time=4ms TTL=127
Reply from Remote: bytes=32 time=7ms TTL=127
Reply from Remote: bytes=32 time=4ms TTL=127
Reply from Remote: bytes=32 time=2ms TTL=127

Ping statistics for Remote:
    Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:
    Minimum = 2ms, Maximum = 7ms, Average = 4ms

To verify that the firewall exception is enabled and allows access:

> test-netconnection Remote -port 5985

ComputerName     : ws-5cd1277hst.ad.ccsd.net
RemoteAddress    : Remote
RemotePort       : 5985
InterfaceAlias   : Ethernet 3
SourceAddress    : Base
TcpTestSucceeded : True

The two computers are in a domain together so the section about the "exception for public profiles" shouldn't apply.

I am able to use the following command to connect:

connect-wsman -computername Remote

When connected this way I am able to explore the wsman directory and see things but I don't really know what I'm looking at/for outside of educated guesses, but it looks like the same stuff that is in the Local Group Policy Edior.

> cd wsman:
PS WSMan:\> ls


   WSManConfig:

ComputerName                                  Type
------------                                  ----
Remote                                        Container
localhost                                     Container

I have also added the option "*" to the trusted hosts on both machines.

What steps am I missing? Is it possible that the school district is preventing this since they are managing the campus network centrally?