Obviously I have zero experience of the bat and how it interacts with avast, for instance there is an Outlook/Exchange provider in the list of avast shields, this too is a plug-in, but one from MS Outlook (not express) which would work in the same way as the bat plug-in.
I just don't know if in the same way there should be a corresponding The Bat provider listed in the on-access scanner window.
Well if you have the plug-in the Internet Mail provider is effectively redundant to you as all email will be intercepted by the bat plug-in, so any thing you set in the Internet Mail provider will have zero impact. Other than if there was something else on your system sending out email outside of the bat, commonly an indication of the presence of a spambot.
You technically you could terminate the Internet Mail provider, but I would suggest you not only leave it but set the sensitivity to High, this would then be able to identify multiple identical emails being sent in a period of time (early indication of an undetected spambot as mentioned above).