If the bridge starts up successfully, but inviting the bot doesn't work and the logs don't show any errors, it usually means the homeserver isn't sending events to the appservice. There are a few potential reasons this can happen:
There was a misconfiguration in the appservice address/hostname/port config or the homeserver can't reach the appservice for some other reason. The homeserver logs should contain errors in this case (grep for transactions in Synapse's homeserver.log).
The bridge was down for longer than a few minutes and the homeserver backed off. The homeserver should retry after some time. If it still doesn't work after an hour or so (exact backoff depends on how long the bridge was down), check the homeserver logs.
Synapse messed up and silently broke the appservice. This is quite rare, but you should check matrix-org/synapse#1834 if nothing else works.