It's not clear whether you're asking about the synchronic phonological analysis or the historical source of the alternation. I would first try to understand what the synchronic pattern is, and then look into the diachronic cause. On that first point, this is basically a standard pattern of phonological analysis, so the usual questions arise. Why do you claim that the underlying form is /patitʰin/ when it is pronounced [patikʰin]? Maybe there are alternations like [patikʰin ~ patitʰan] where -in, -an are suffixes. Why do you assume /patitʰ/ and not /patikʰ/? Maybe there are roots with non-alternating final /kʰ/ but no roots with non-alternating final /tʰ/. Does the sound have to be aspirated, or are there parallel alternations for /t,d,s,z,n,l,r/? Why doesn't /t/ in the second syllable change? Maybe because it only happens in the last syllable of a word. Does it happen with any other affixes or just one? And so on. On other words, there are lots of facts about such an alternation that we would have to know in order to be able to say what the rule is.
Suppose that you have answers to enough of these synchronic questions and your interest is, what was the historical change involved. This is harder to answer, because not only do you have to do a synchronic analysis of this language, you have to do a similar analysis of related languages. This is remniscent of Carpathian Ukrainian dialects, where palatalized /t,d/ become [c,ɟ]. You might find that, if the apparent trigger is /i/, that vowel triggered palatalization and there was a change /tʲ/ → [c]. The connection to aspiration is that aspirated consonants have a noisy release, and vowels have their strongest anticipatory effect on noisy releases, so aspirates are more likely to be affected by the following vowel.
The change from tʲ to c is a compression of the reverse of the kʲ to tʃ change. The acoustics of tʲ and c are quite similar, so people hearing (intended) tʲ could analyze what they hear as c, which is virtually the same as kʲ. Of course, if you do not have any information on earlier stages of the language or on related languages, you can't test the specific historical scenario were convincingly.
I would start the analysis by getting a firm understanding of what the contemporary facts are, and then try to discover the causal mechanisms behind those facts.