You are still a little confused about Q. Q is not 'true' or 'false', Q is observed or not-observed. Q is not an explanation that can be true or false. P is the explanation and can be either true or false.
"How else is P to be supported?"
P is supported by eliminating all alternatives. Otherwise you are simply engaging in logical fallacy. That's the point.
"How else does one investigate a theory or idea than to test whether the predictions (Qs) of the theory/idea/claim (P) are true or not?"
What you are beginning to realize are the limits of science because theories are dependent on logical fallacy. This is why theories are constantly overturned and is an inherent weakness of science. If your theory is philosophical (like evolution) you cannot claim that it is scientific because huge portions of it are assumed, untestable and unfalsifiable. If your theory is methodological (like superconductivity) you may claim that it is scientific but it is still subject to being thrown out because not-Q can be observed at any time.
In sum, methodological theories can be overturned at any time and philosophical theories are unfalsifiable.
"—The fallacy states: “If P, then Q. Q is true. Therefore P.” In the language of logic “therefore P” means “a logical certainty” or “absolutely proven”, not merely “supported”."
It is not a 'logical certainty' or 'absolutely proven' that P is supported. It may appear that P is supported but alternatives to P may be supported as well. P may very well be false and Q may not have supported a false theory at all. That's the fallacy.
“P is supported by eliminating all alternatives. Otherwise you are simply engaging in logical fallacy. That’s the point.”
—No, P would be PROVEN by eliminating all alternatives (a feat that’s unrealistic). Why do you equate PROVEN with mere SUPPORT?!
“What you are beginning to realize are the limits of science because theories are dependent on logical fallacy.”
—Yes, science is limited – but not because theories can’t be SUPPORTED, but because they can’t be PROVEN.
The fallacy is in equating the two – which is what you are doing – although in a new bizarre way. The fallacy tells us that although theories can be “supported” that they can’t be “proven” – and thus warns against equating the two. You likewise equate the two by saying that since theories can’t be proven that they thus can’t be supported! (This may be the discovery of a new fallacy – I wonder if you get to name it? :-) ) Which would mean all science is a fallacy (which at least you admit “theories are dependent on logical fallacy”).
“It is not a ‘logical certainty’ or ‘absolutely proven’ that P is supported. It may appear that P is supported but alternatives to P may be supported as well.”
—Yes, that’s one of the differences between “supported” and “proven”. Multiple competing theories may be supported by a finding, but of course multiple competing theories can’t be proven true.