You don't need a lockdown browser for this. You can block the global internet on Windows (or Linux, but I presume you use Windows) via firewall rules.

Search for "block all sites except one", this is a common practice. Here is one video. Leave only the LMS accessible. Here is the option with Tinywall or older Windows firewall.

Then, give students a limited parental-control account for test-taking, where everything that could be used to cheat is blocked.

A very small proportion of students will be able to bypass the blocks, but you probably shouldn't worry about their test scores in an offline setting.

If you have admin rights, you don't need a lockdown browser for this. You can block the global internet on Windows (or Linux, but I presume you use Windows) via firewall rules.

Search for "block all sites except one", this is a common practice. Here is one video. Leave only the LMS accessible. Here is the option with Tinywall or older Windows firewall.

Then, give students a limited parental-control account for test-taking, where everything that could be used to cheat is blocked.

A very small proportion of students will be able to bypass the blocks, but you probably shouldn't worry about their test scores, as it takes pretty good CS skills.

Ideally, you'd ask your systems administrator to do this via group policies, it's an easy task for any full-time admin, commonly done in organizations.