I'm having a little trouble understanding what you mean when you say that your name should contain two "t" sounds in a row.
At least in English, I don't think it's possible to have a word where two individual "t"s are pronounced in a row. A double "t" in English is pronounced as identical to a single "t" when it occurs in the middle of a word (eg. in the words "mitten" and "batter"), and when it crosses a word boundary it's pronounced as a "lengthened consonant" just like those used in Japanese (with a pause in between the consonant's onset and release, eg. in the phrases "put together" or "night train").
I believe Puttichai is a Thai name (correct me if I'm wrong). I'm not at all familiar with Thai phonology myself, but Wikipedia indicates that plosives in the Thai language are always unreleased when they appear at the end of a syllable. This suggests to me that perhaps the double "t" in "Puttichai" should be an unreleased "t" followed by a released "t", which I think is functionally identical to a "lengthened t". If this is the correct pronunciation, I think プッティ would be a very close representation, much more so than プティ (which is unambiguously a single short "t" sound).
As for the latter half of the name, if your phonetic representation ʃaɪ is correct, this would indeed be better represented as シャイ (pronounced the same as "shy" in English) rather than チャイ (pronounced the same as the first syllable of "China" in English).