Is this what you are trying to do?
$ docker run --add-host foo:127.0.0.42 alpine ping foo -c 3
PING foo (127.0.0.42): 56 data bytes
64 bytes from 127.0.0.42: seq=0 ttl=64 time=0.045 ms
64 bytes from 127.0.0.42: seq=1 ttl=64 time=0.061 ms
64 bytes from 127.0.0.42: seq=2 ttl=64 time=0.070 ms
--- foo ping statistics ---
3 packets transmitted, 3 packets ...
If I'm getting this right; Maybe you can just create on your old account's Route53 public hosted zone a CNAME record (if you have a load balancer on your new account) or an A record for the instance on your new account. Just don't use the Alias functionality from Route53 as you wont find the resources from the new account.