If you can get the museums to make a change to the browser configuration you can issue certs without the involvement of an external CA: you can make your own CA for your application (perhaps just by using a few openssl command line commands, among other possibilities) and then ask the museums to add the CA's root certificate to the kiosk browsers. Then the browsers will accept the certs that you issue, without needing to involve any outside party.
If you don't want the museums to have to make a configuration change in their browsers, you could get the certificate from Let's Encrypt for the name under myapp.com, as you describe, and then deploy it in the museum. The one difficulty about the redirection is that the browsers will need to make an initial HTTP (not HTTPS) connection to the captive portal in order to get redirected to the site. That is probably not a problem. It does mean that the level of security you would get against an attacker who gets onto the network is lower, because the attacker might be able to intercept the initial connection and redirect to the attacker's machine instead of to your server.
I think the internal CA is probably more appropriate for your application overall, and many organizations do something like that pretty successfully. Then you don't have to rely on Let's Encrypt's policies, mechanisms, or technology, and you can even use the names of your choice (not only names under myapp.com).