I was able to dig up an actual copy of the referenced Letter 153 and the text of the letter makes many things clear.
1. Túor becoming Immortal is only a supposition, and not fact.
... my legendarium, ... is based on my view: that Men are essentially mortal and must not try to become 'immortal' in the flesh. (Since 'mortality' is thus represented as a special gift of God to the Second Race of the Children (the Eruhíni, the Children of the One God) and not a punishment for a Fall...)
... Túor weds Idril the daughter of Turgon King of Gondolin; and 'it is supposed' (not stated) that he as an unique exception receives the Elvish limited 'immortality': an exception either way.
So, Túor becoming immortal could be true, but could also just be something Noldor in Middle-earth believed without definitive proof of truth, as per Tolkien's emphasis. Since this information is supposed to be coming to us from the beliefs of the Noldor stuck in Middle-earth and the trip to Valinor is a one way trip, accurate knowledge transfer from Valinor to Middle-earth presumably doesn't happen.
If the myth is, in fact, untrue, one source of this belief could be the rare case of Glorfindel's return to Middle-earth. One would have to make multiple assumptions (fortunately, all of them things debated but not settled by Tolkien himself) - that Glorfindel of Rivendell was indeed a re-embodied re-incarnation of Glorfindel of Gondolin; that Glorfindel returned to Middle-earth sometime in the Second Age; and that while Aman does not stop mortals from dying, it can extend their life to the fullest, so that Túor was indeed still alive at the point when Glorfindel left the Blessed Lands - and that this was the piece of news that gave rise to the idea of an immortal Túor amongst the Noldor Eldar and Edain in Middle-earth.
2. Lúthien's exception is referred to in similar terms as Túor's exception
In the primary story of Lúthien and Beren, Lúthien is allowed as an absolute exception to divest herself of 'immortality' and become 'mortal' — but when Beren is slain by the Wolf-warden of the Gates of Hell, Lúthien obtains a brief respite in which they both return to Middle-earth 'alive'.
Though of course, Lúthien's fate is considered more 'factual' than Túor's. Presumably the difference is that Lúthien's death, return and final death was something the Noldor in Middle-earth could corroborate - and if Glorfindel could share news of the life of Túor, he could certainly convey the news that Lúthien had not returned to the Halls of Mandos after her demise.
3. Tolkien answers "As for 'whose authority decides these things?'"
The immediate 'authorities' are the Valar (the Powers or Authorities): the 'gods'. But they are only created spirits – of high angelic order we should say, with their attendant lesser angels – reverend, therefore, but not worshipful; and though potently 'subcreative', and resident on Earth to which they are bound by love, having assisted in its making and ordering, they cannot by their own will alter any fundamental provision.
Immortality and Mortality being the special gifts of God to the Eruhini (in whose conception and creation the Valar had no part at all) it must be assumed that no alteration of their fundamental kind could be effected by the Valar even in one case: the cases of Lúthien (and Túor) and the position of their descendants was a direct act of God.
4. The Reason for Lúthien and Beren being allowed to come back to life is given
Eärendil is Túor's son & father of Elros (First King of Númenor) and Elrond, their mother being Elwing daughter of Dior, son of Beren and Lúthien: so the problem of the Half-elven becomes united in one line.... The view is that the Half-elven have a power of (irrevocable) choice, which may be delayed but not permanently, which kin's fate they will share.
... The entering into Men of the Elven-strain is indeed represented as part of a Divine Plan for the ennoblement of the Human Race, from the beginning destined to replace the Elves.
It doesn't definitely answer why Túor and Lúthien were given special dispensation to join their spouses' ultimate Doom (and why the solution was different for both couples), but we can assume this was as reward for folks whose love (and resultant procreation) was an important part of the Divine Plan, instead of the 'punishment' that eternal sundering might have been to them.
Rob answers the 'Why was Beren in the Halls?' part of things very correctly, and also explains how Manwë was able to confer the choices to Lúthien. The spirits of Men seem to make pitstop in Mandos before leaving the confines of the World as is their Doom.
It was to the Halls of Mandos that the spirits of Elves and Men were gathered to await their different fates**, and so Mandos was given its name of the Halls of Awaiting. After a time, the immortal Elves could be re-embodied, and return from the Halls to their kin in Aman. Men had a different fate, a fate which, even among the Lords of Valinor, only Mandos and Manwë truly understood. No one, however, not even Morgoth could escape the Halls without Mandos' permission. [wiki src]
Beren extended his stay on Lúthien's request:
"For the spirit of Beren at her bidding tarried in the halls of Mandos, unwilling to leave the world, until Lúthien came to say her last farewell upon the dim shores of the Outer Sea, whence Men that die set out never to return."
And Mandos was able to gain counsel on what to do from Ilúvatar himself.
"Manwë sought counsel in his inmost thought, where the will of Ilúvatar was revealed."