If one doubts, in the last rakʿah, whether it is the third or the fourth, and initially ignores the doubt, then makes takbīr to stand for the fourth rakʿah during the final tashahhud, does this necessitate a pre-salaam or post-salaam prostration of forgetfulness (sujūd al-sahw), and is this extra takbīr during the tashahhud considered an addition to the prayer?

If one experiences frequent doubts, to the extent that they occur every day, even if only once, then they should disregard these doubts and not act upon what they doubt. In such a case, the prostration of forgetfulness (sujud sahv) is not obligatory. If you doubt whether you have prayed three or four rak'ahs, you should consider it four and not perform what you are unsure about. Your performing the fourth rak'ah due to doubt does not invalidate your prayer. It would have been correct to stand for the fourth rak'ah without an additional takbir, as the previous takbir would suffice. Your sitting for the tashahhud in the wrong place and your takbir when standing for the fourth rak'ah are considered additions for which the prostration of forgetfulness is prescribed after the salutation (taslim). The post-salutation prostration (sujud ba'di) is valid if performed before the salutation, just as the pre-salutation prostration (sujud qabli) is valid if delayed until after the salutation.