It is very much possible to ticket LHR-IAH-BZE (and return) all on UA and via IAH only. The issue might be due to IAH to BZE operating the next day (morning flights only), so UA may be treating it as a stop over (instead of a layover)...will have to review the fare rules in greater details.
A competent travel agent can book this fare for you even it might not be visible on UA.com - assuming in economy, you can feed the TA this fare code: VMPMB4BV
Is there a specific reason to book UA? DL/VS via ATL is very much bookable online!