Another good website is www.findmypast.ie
It shows many Handibodes in Westmeath, Offally and Dublin.
This is a subscription site but it also has a "pay as go" facility. It also offers a 14 day free trial which would give you time to download all the Handibode data.
The border between Westmeath and Offally is right in middle of Horseleap- Westmeath to the East and Offally to the west. It is likely that this area of Offaly would have Tullamore as its B, M and D registration location.
Hope this helps in your search.