[RPG] Godly non-interference and godly summoning [Rise of Tiamat spoilers]

This answer reads in four sections:

1. Background: where some of the possible inconsistencies from the designers are described and a warning that a definitive answer that is satisfactory for everyone might be absent. The specific examples given relate directly to Tiamat; and links to further auxiliary information, which might be useful in setting a background while running the module, are provided.
2. Answer from Lore: where the overall premise of the importance of leaving the prime material to the dealings of the mortals is described, using data from earlier editions of D&D. Given that the default cosmology of the 5e is very similar to the 2e, the arguments in this section are likely to be still valid.
3. Regarding Bahamut: where clues are provided from the 5e Monster Manual about the behaviour patterns of Bahamut, the one deity beyond Tiamat that is explicitly mentioned in the question.
4. 5e Forgotten Realms Specific: where we argue that the tone of the Realms past the Second Sundering is one with deities still casually around, but are "subtly" leaving the fate of the world to the mortals.

Background

The answer to this question appears to be tricky, as there is a need for the players (and hence mortals) to be at the center of the gaming world, but at the same time there is a desire to design ever more challenging adventures for them (stopping evil powers from invading your home world is challenging). Moreover, Forgotten Realms being a very large setting that has had a special standing since the days of TSR, with many authors and game designers contributing to it, there are occasional lapses in continuity, which are retconned later.

For example, over the many editions of D&D, Tiamat had realms/residences in various Outer Planes, became a servant to Bane and Asmodeus, had three different aspects, and finally became to be the subject of the module Rise of Tiamat. Since this created confusion, folks at candlekeep.com asked the opinions of Ed Greenwood, the originator of the Forgotten Realms setting, and also the first person to have described the Nine Hells within D&D context (in two articles that appeared in the Dragon Magazine in 1983, articles in which he had mentioned Tiamat as the ruler of the uppermost layer, Avernus). You can read his full answer in a post from Feb 2015 on Candlekeep.com; it is a long answer mentioning the aspect of Tiamat from Rise of Tiamat explicitly at some point, so I suggest you have a look at it to put that module into context. The short part I want to quote is about how even the most knowledgeable designers, even Wolfgang Baur of Kobold Press, who published the module itself, could not see the full picture:

So why does she need rescuing from Avernus in the Tyranny of Dragons storyline? I briefly chatted to Wolfgang Baur about this but all he said was that "It was the premise provided by Wizards, and Kobold Press really isn’t in a position to dictate canon to the Wizards team."

Ed spent some time trying to get an official answer for you, and at length abandoned that attempt and spun his own (which of course he has the moral and legal authority to do, although Wizards can of course at some future time supercede what I’m posting below).

In short, it is possible that the designers or the product commissioners might have missed thinking about the serious issue raised in this rpg.se question.

Answer from Lore

The 2e Planescape product On Hallowed Ground, is a 192-page book dedicated to deities and their servants in all campaign worlds of the AD&D game in 1996 (which included Forgotten Realms). In it we read (page 40):

The Prime Material Plane is a major point of contention among the powers. ... The primes may be clueless, but they have faith. They believe without having to see, and that's a rare gift - especially on the Outer Planes, where belief is made flesh. The gods ... know that belief is power, and they see sustenance ... in the uptapped faith of the primes.

That's why the powers don't allow each other directly on the Prime, why they have priests and proxies, and why they wage holy wars against each other. 'Course, the gods don't always stop to think: If one pantheon dominates and the rest die out, might not the winning powers eventually fade away too? Perhaps the gods feed off one another as much as they feed off mortals.

Just about all the powers in the multiverse have a stake in the Prime, and they don't want to mess up such valuable ground. ... Avatars, ..., aren't strong enough to cause such devastation. The gods agreed to make their avatars weak enough to be taken out by a very powerful mortal. ... Simply put, gods who start warring on the Prime are just begging to have every other power in the multiverse eliminate them.

Hence, it seems like even when a crazed deity like Tiamat wants to enter the prime material plane, she does not enter on her own and need to be "invited". While other deities would not like that to happen, they need to trust their own servants; mortal heroes can be as effective as their avatars. This is the way one needs to abide by if one does not want an arms race that will ruin the prime material plane.

Regarding Bahamut

Since the question mentions Bahamut in particular, it is worth mentioning that there is a clue in the 5e Monster Manual (page 103):

Bahamut ... dwells in the Seven Heavens of Mount Celestia, but often wanders the Material Plane in the magical guise of a venerable human male in peasant robes. ... Bahamut seldom interferes in the affairs of mortal creatures, though he makes exceptions to help thwart the machinations of Tiamat the Dragon Queen and her evil brood.

Perhaps Bahamut is already playing a role behind the scenes, but being a deity of law, he might be holding onto the rule/agreement of not interfering directly, realising that in the very long run, this would harm the prime material. Besides in the grand scheme of things, who can say that a new "age of dragons" would be a bad thing for Bahamut in the long run? As the evil dragons start ruling the lands, there is a chance that there will be more worshippers who would appreciate the Platinum Dragon.

5e Forgotten Realms Specific

Finally, with the rewriting of the Tablets of Fate by overgod Ao, the tone of the Forgotten Realms past the Second Sundering is claimed to have somewhat changed. According to the Sword Coast Adventurer's Guide (page 18):

By 1489, many of the wars that began during the Sundering had ground to a close. Other conflicts arose, and mighty threats still imperiled the world, but the deities ceased interfering with the world through their Chosen. The gods were no longer silent but quiet, and in many places new priesthoods arose to interpret the gods' now subtle signs.

How subtle the signs are and what quiet really means is difficult to assess, given the scale of the disaster that needs to be avoided at the end of Rise of Tiamat. Different contributors to the setting seem to have somewhat different takes on this. For example, the article about Eilistraee on the FR wiki refers to the novel Death Masks by Ed Greenwood and states:

Eilistraee was seen dancing and speaking to mortals in many places, especially along the Sword Coast. For example, the citizens of Waterdeep witnessed the Dark Dancer, as she danced in the moonlight, near the walls of the city, up the road to Amphail.

So seemingly casual dancing and talking to mortals is possible for at least some deities. Perhaps they are "subtle" in the sense that the avatars, like the venerable human avatar of Bahamut, are just talking now, and expecting that the mortals take the action. This interpretation appears mostly in line with the description provided by Matt Sernett and Chris Perkins in the Dragon Talk podcast from December 2016 (episode 'David Eddings on Roleplaying Claptrap', after the 8th minute):

the Sundering is really a statement about where we want to take the Realms going forward, which is: the gods are around but at the end of the Sundering they withdraw from the world to an extent and basically usher in what could be considered an age or mortals, where the fate and future of the world will be decided by the people in it, not by the gods.