Why is Mises "permitted" to define "Economics is an a priori science", if a priori is not trustworthy? Why is Mises "permitted" to define "Economics is an a priori science", if a priori is not trustworthy?

In philosophy of science and natural sciences a priori as a method of knowing has long been identified to be sufficient only in special cases and more broadly mainly just among the formal sciences (mathematics, ...). By sufficiency I mean that it's sufficient for obtaining enough information in order to make empirically accurate predictions.

Since economics is about empirical things (people, resources etc.), then defining economics as an a priori science just seems very naive, if not even dangerous, because when combined with politics, such economics could propose anything and then claim validity to it, because "well it's an a priori science". What if one claimed that "Islam is an a priori science"?
Although I disagree with von Mises, he was reacting to what he saw as a very real problem, which was the use of relatively primitive statistical methods to describe behavior.  In fact, Leonard Jimmie Savage's work on probability and statistics should have been sufficient to meet his objections, although the computational speed would not have yet existed for that to have been worthwhile.

The challenge the Austrians have is that they are really arguing that no regularities exist in human behavior.  They do not see it that way, but probability and statistics are about the discovery of regularities.  Their position is that humans are too unique to be subject to statistical measurement.  All of economics could be solved axiomatically, in his understanding.

I would agree with him, as long as you got every axiom correct and the set was complete and properly partitioned for all possible cases.  Of course, the only way you could know that would be empirically and that would defeat his argument.

Although other branches of economics use axiomatic assertions, they then check to see if the predicted behavior really happens that way.  If so the axioms are at least not disconfirmed.  Most assumptions are also things nobody would disagree with.

For example, there is a proposition that humans prefer some things to other things.  If you also assume that for any pairing of goods, we will call them x and y, then either y is not dispreferred to x or x is not dispreferred to y, or neither or dispreferred to each other. If you add transitivity where for all triples x,y,z then if x is not dispreferred to y and y is not dispreferred to z then z is not dispreferred to x.

If those conditions hold, then it can be shown by theorem that it must be true that a utility function exists such that if the utility of x is not less than the utility of y then y is not dispreferred to x.

If those assumptions do not hold, then something else is true, but if they do hold then the entire mathematical theory of functions opens up to economics.  With the addition of some other mild assumptions, then calculus becomes available.

These do not need empirical checking because it is the only possible outcome, by force of math.  To go much past this point, however, becomes tenuous.  For example, must some of these functions be concave or are there circumstances where they could be convex.  If they can be convex, do real-world examples of this convexity exist?  Do real-world examples of the concave case exist?  Are there forced implications from either case or the existence of both cases?

These questions can only be approached empirically.