A number of RC parishes in Co. Westmeath do not have records back to 1826 so we don't know if a record for Eliza is available. I did search on the subscription site Roots Ireland and found a Hegarty family in the Collinstown RC parish in Co. Westmeath which may be a good lead.
The parents were Edward Hegarty and Mary Neill. The children were Eleanor 1809 Catherine 1811 Daniel 1813 Mary 1821 William 1823 Ellen 1826 Margaret 1831 NOTE there is a gap in Collinstown RC baptismal records from 1815-1821 so maybe more children. Also, the mother for the 1831 Margaret was shown as Margaret not Mary.
Here is a link to the parish register https://registers.nli.ie/parishes/0918
Have you considered DNA testing?
Let me know what questions you have.