To be sure you are tracking the correct Thomas Sullivan, you should first obtain the full copy of the Birth Record for Christopher Gilbert Sullivan and use the details on that to follow up through CG's father, William.
These full records can be obtained from the GRO Southport Lancashire, by following the link on Find My Past.
They cost around £10 each and take about two weeks to be delivered.
After confirming Thomas that way, you should obtain similar records for him, using the details on the other, later generation, records, as otherwise you may be wasting money.
Assumimg that you do properly identify Thomas; and that he is from Ireland, you then have to make similar searches etc onn the Irish sites, mainly-