LWP::UserAgent is extremely well established, so yes, you could just stick with that if you figure out the correct options*. Other than that, I'd recommend HTTP::Tiny, or, if you want to step it up a bit, Mojo::UserAgent, which has the advantage that it gives you easy access to JSON and XML/HTML parsing features via Mojo::JSON ($ua->get($url)->result->json) and Mojo::DOM ($ua->get($url)->result->dom).

* LWP is not my area of expertise, and you also don't say exactly in which way the requests are going wrong (SSCCE, How do I post a question effectively?). Tracing a good request and comparing with a bad one using Wireshark might be very useful here.