Is that system accessible to the public internet? Because I don't see any A or AAAA records in the DNS so I am thinking it may be on a private network. An HTTP Challenge will require that name to reply to HTTP queries from the public internet.
Do you have update authority to the Route53 DNS? Because you can get a cert using a DNS Challenge but you need to be able to update the DNS records.
The Getting Started section is helpful if you have not yet reviewed it: Documentation - Let's Encrypt
Lastly, your Route53 DNS name servers have a likely problem. This should be fixed regardless of whether you continue with Let's Encrypt. I mention this only as I saw it while researching your question.
You need to make sure that the 4 nameservers listed under "Hosted zone details" in the Route 53 console are the same ones listed at both your registrar, and in the NS record for your zone. See "Step 4" of the AWS Docs here: Making Route 53 the DNS service for an inactive domain - Amazon Route 53
See this DNSviz report for the error
https://dnsviz.net/d/acnixiam.com/dnssec/