Does prayer become invalid in the event of doubting the number of rak'ahs, then adding to them as a precaution, with the intention of performing the prostration of forgetfulness (sujood as-sahw), and upon tasleem remembering the prostration of forgetfulness, returning to it, and then performing tasleem?

What you did by building on the minimum, performing a fourth رکعة, and prostrating for forgetfulness (Sujud as-Sahw) is correct. However, you made a mistake by returning to Sujud as-Sahw after you had already begun the tasleem (salutation to end the prayer), because you returned from a pillar (rukun) to a sunnah. If this return was intentional, then the prayer is invalidated. But if it was due to ignorance of the ruling, then your prayer is valid.