Incorporate and apply for EIN through a foreign entity
As far as I understand this is not a problem. The company will require an EIN number to operate. I do not want to provide any personal data for the formation of the new Texas LLC.
My question: Can the UK Ltd file the SS-4 form without having a TIN or will the UK Ltd be required to apply for a TIN. Do you have any other suggestions?
Theoretically it should not be a problem. You would still need to sign the SS-4 as the authorized representative of the UK company, but no other personal information would be needed. The UK company might need to obtain its own EIN eventually in order to pay taxes on the income in the US (derived from the US LLC that it owns).
In practice it would depend on IRS examiner who would work on your SS-4 application. When we call the IRS sometimes the examiner chooses to reject applications such as the one you described, sometimes the same application would be accepted (thats why we call several times until we succeed).