Do you want to do the course in sidemount or ccr, TDI or IANTD or nss cds is the way to go.
You want to use a stage already in the course: IANTD, with TDI there is no stage used. Both agencies have the multiple stage cave course, but if you are a trimix diver, you already know how stages work and the gas rules are no rocket science. If you think I want that course also, take it. This depends on your other stage diving experience.
If you are ready for it, and is time also important: TDI or IANTD. You can do it in 1 week. Price around 1500-2000 dollar. This can be a good choice, but is not for everybody.
Intro to tech, fundies, Essentials: not for every diver needed. It is not a requirement for TDI and IANTD or CDS. But a good instructor will do an evaluation dive and look if you are ready for tech or not. This can save you a lot of money and time. If you need it, do such a course. You need to have a minimum level before entering a tech course. But some can achieve this level by talent, willing and the right buddies and being an autodidact. The evaluation dive is then more or less an exam to look if you are ready or not. Having passed such a class doesn't say a lot if you don't practise the skills after it. You must stay current. Sometimes I have to advice to practise at home first before entering a cave class.
But the most important thing: Does the instructors personality fits you? You can go to the best instructor, but if the personality does not fit, it is not the best instructor for you.
Another important thing: how are the courses teached? Some instructors have a rigid way because of standards or personality. Others are less rigid. For example: I don't teach a course with only theory on 1 day, on every course day there is at least 1 dive. I will or would never take a course with just theory and no diving on a course day. Within TDI and IANTD I know you have such freedom. Of course you cannot jump in a cave and start a cave dive without some theory or landdrills, but to do all theory in the first day and no diving is really really boring and will not help as information will get one ear in, other out. Another important thing: is there time to have some discussions? Only do yes and amen is not the best way to learn, but some instructors only teach this way. If you can have the yes, BUT discussion, then I would prefer this.
The instructor is more important than the agency.
I did my cavern, intro and full cave in 7 days, 16 dives. It was a good course with 1 of the Thailand cave rescuers (now more or less wellknown, in 2012 'unknown'). But I already was an adv. rec. trimix diver when entering the course, so already had 'some tech' experience. It was a good course. The reason to do it in 1 week was for me: I don't live near caves and want to try to get most out of 1 trip. And I passed. As I had to travel, I planned some extra days, in case when needed I could use them and still going home with a cert. We started every day with some theory in the morning, then do 2-3 dives.