I think Flying Thunder God should work considering:
1. All Space-Time jutsu work on a similar principal.
2. Using a Space-Time jutsu to leave a dimension has been done (Obito's use of Kamui to access Kaguya's Dimensions, Edo Tensei summoning souls from the Pure Land, Sasuke's use of his jutsu to move from Kaguya's Dimensions and back, the use of Space-Time Ninjutsu to move from Momoshiki's planet and back, Sasuke was able to teleport Madara's Limbo Clone (which reside in another dimension) and Hagoromo's summoning of both the dead and alive from the Pure Land and Kaguya's Dimensions. So there are plenty of examples of this being done.
3. There hasn't been any evidence suggesting it wouldn't work. If anything, at most it seems to me that it would cost more chakra on part of the user to move things across dimensions (Sasuke's eyes weakening, the chakra toll on Obito, Kaguya's own use of her jutsu, etc.)
Using a space-time ninjutsu to travel to another dimension has been done, yes, but none of those jutsu were Hiraishin.
Your third point is an argument from ignorance. You can't make up a conclusion that it would work, albeit at the cost of greater chakra, simply because there's no evidence to the contrary.
In fact, absence of evidence, which is the failure to observe evidence for a particular hypothesis (in this case the hypothesis that "using hiraishin to travel to another dimension is possible, albeit at a greater chakra cost"), is evidence against that hypothesis.
But why would Hiraishin be the special case if other space-time jutsu work in a similar manner? (I think as long as you have a seal placed in our world, one could make their way back) We would have to have evidence that Kamui has a sort of special Space-Time negation ability.
Its speculation sure, but I backed it up with support (what exactly the series shows happens when a Space-Time jutsu is used to escape another dimension almost every single time). Isn't equally ignorant to assume since we haven't seen Flying Thunder God do it, that it cannot? We need to look at the series itself and make a judgement on it, and to me, Kamui doesn't seem like it would stop this process for any particular reason.
Based on the observed trend (that most space-time ninjutsu allow for interdimensional travel), it's reasonable to suspect that hiraishin would allow you to do the same. However, making that assumption without due evidence is wrong because as empirical studies often show, there can be exceptions to the rule.
Like you said, when need to go off of the evidence the show provides, and it has provided no evidence that hiraishin allows for interdimensional travel.
In order to point out the flawed reasoning of your question, I have to ask you this: Is it ignorant to assume that since we've never seen a horse fly, that they therefore cannot?
Also bringing up the kamui thing is a separate issue, since I've yet to see anyone on this thread assert that kamui somehow stops hiraishin from moving between dimensions - simply that hiraishin itself cannot.
What exception? Without said exception even present, its simply a matter of "what you want" rather than "is it reasonably possible". Until proven otherwise, to stick such a limitation on the jutsu in unfounded.
That horse fly analogy isn't even close to this. Even if we never saw a horse fly, we know based on evidence that it lacks the capabilities to do so. If this argument was valid, there would have to be evidence that Flying Thunder God only allows for travel in one dimension, which there is none.
Actually Shadow brought up his reason as to why he believes Kamui prevents Flying Thunder God. So I put it out there to ask what exactly is so different about other dimensions such as Kamui, that would prevent a user of Flying Thunder God to manipulate the Space-Time continuum? I just think, based on the use of other Space-Time jutsu that has preformed this feat (another example is Sasuke summoning his Hawk into Kaguya's dimension), that it is far more reasonable to say, "yes it can" rather than "no it cannot". Even though Hiraishin itself hasn't been shown to do that, to me, its like assuming "fire cannot burn down a building because i've only seen it burn paper all my life." Well logically, unless there is something special about the building (in this case other dimensions like Kamui) fire should still it burn regardless.
Lol, the exception to this case (that jikukan ninjutsu allow for interdimensional travel) is hiraishin, since there is no evidence showing it does.
The horse analogy is exactly the same argument that was made. All I have to say is that there's no evidence against the fact that horses can't just fly through some other means (since they clearly lack the anatomical equipment that is necessary for flight). "Well theres no evidence against the fact that horses fly using fairy dust when no one is looking, therefore it's true!" This is why arguments that assert that something is true based on a lack of evidence to the contrary are flawed. I don't see what's hard to understand about that.
Here's another example: Russell 's Teapot. "If I were to suggest that between the earth and mars they're is a china teapot revolving around the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is an intolerable presumption on the part of human reason to doubt it, I should be thought to be talking nonsense." - Bertrand Russell
Basically, even though it makes sense, to conclude that hiraishin allows for interdimensional travel is not only a hypothesis that lacks evidence from the show, it is an unfalsifiable one (since there is no way for us to disprove this hypothesis save for asking Kishimoto himself). That's why you shouldn't make that conclusion.
That's also a bad analogy because the only reason that conclusion is logical to us is because we have mountains of evidence that shows that fire can burn buildings, and we therefore use inductive reasoning to reach that conclusion.
If you were the person who had only seen fire burn paper, then that wouldn't be a logical conclusion to you at all.