Accessing local/Windows Authenticated SQL server over network

Question

I'm still very green with SQL, but I need to learn as much as I can for a couple projects at work. We use AlarmLock, an access control system that utilizes an SQLExpress database on a Windows machine to keep track of employees and their door codes. I have an instance running properly on a Windows machine, but need my ARM64 laptop to connect to the database over local network so I can physically connect to the lock and program it using data from the SQL server (I can't install MSSQL on ARM64 hardware, or this would be much simpler). The database is automatically set up during the DL-Windows (AlarmLock) installation, so I have no control over credentials or authentication type during the installation process. I can see the server running in the host machine via the SQL Server Configuration Manager, and can access the database using Azure Data Studio, but the remote machine refuses to connect, likely for authentication reasons. The host uses Windows Authentication (which I gather to just be "MACHINENAME\USERNAME" & null password), but any connecting machine needs to use SQL Server Authentication to connect over the network, right? The laptop can see the server instance (BP-WS-CONTROL\ALSQLEXPRESS), but can't connect. I have tried switching the "Log on as:" to "This Account" followed by my DOMAIN\admincreds (and yes, I did use "check names"), but this doesn't seem to work. How do I change or add the Authentication to accept inbound connections? I'll add a couple screenshots so you have an idea of what I'm dealing with. I have attempted to adjust some settings to achieve my goal, but I wound up with one machine refusing to connect to even its own database, so I've turned to you before I wind up bricking another machines database. I'd be happy to provide any additional info, screenshots, etc.

Host Machine with correct settings for local access:

Laptop which can't connect:

I hope I've given enough information. I'm used to Python, but SQL is a bit confusing to me so far. Thank you for any help!