(1) Such a scenario has never occurred on Earth. Despite a truly enormous variety of environments and ecological niches on Earth, no prokaryotic population has ever been discovered that has completely re-invented entire suites of core molecular machines (ATP synthases, transcription machinery, ABC transporters, etc.). That this has not occurred suggests that the core molecular machinery of biological life is not so easily replaced by evolutionary processes, regardless of the environment.
Once one type of prokaryotic population became dominate, wouldn't it wipe out less competitive alternatives, and subsequently soak up all primitive organic molecules from which primitive life develops?
(2) The scenario would not be believable from a molecular evolutionary perspective. If the hypothetical FUCA -- containing a wide range of radically different core protein sets -- could survive on Earth upon landing, then these protein sets would most likely be tinkered with and gradually co-opted and improved upon, instead of massive deletions taking place along with massive molecular innovations.
Wouldn't they be treated as food?
(3) A large number of these core proteins function in an information processing context and so are not immediately affected by the external environment. E.g., polymerases are involved with nucleic acid polymers and do not interact with the external environment, so postulating a radically different external environment for the "source planet" would not require a radically different set of information processing proteins. These proteins would arguably retain their basic tertiary and quaternary structure regardless of the external environment.
Wouldn't polymerases be digested as food instead of incorporated into existing life?
(1) Molecular clock analyses. If the core protein machinery of prokaryotic life was the product of radical evolutionary innovation, then molecular clock analyses should place the time of origin of core protein machines at around 3.5 Ga, in concordance with paleontological evidence for when membrane-bound prokaryotes emerged on Earth. This is not the case, however, so this conjecture is not consistent with molecular phylogenetic evidence.
Do you have references showing that "molecular clock analyses should place the time of origin of core protein machines at around 3.5 Ga?" And since one of the most common headlines in anthropology, paleontology and abiogenesis research is, "Scientists push back earliest date of <fill-in-the-blank>," that we've found nothing before 3.5 Ga isn't particularly meaningful.