If he was a foundling with only a name tag, saying his name was "Frank Elliot", how do you know he was related to the "Elliots of Birtley, musical family?" I find it a little odd that a child left on someone's doorstep would have been named?
If his given name was Frank you also might like to consider searching for him as "Frances".
What is the source for the information you have about Frank being a foundling?
Unless he was named in a will as illegitmate son of said Lord, I'm not sure if it will be possible to prove that story. I think lots of stories were concocted to cover illegitimacy - and often family stories tend to play up this kind of thing. It sounds mrore acceptable to say an illegitimate ancestor was a product of the local Gentry and a servant than the product of a roll in the hay with a farm boy. (Though of course both are possible.
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It could be worth investigating DNA testing.
It could also be worth considering that Frank was the child of the Taylors, but for some reason changed his surname to Elliot?
These are just a few thoughts from a superficial reading of your question, and without looking further into the family, so may or may not be relevant.
