Just for those who might infer the contrary, being a T on the MBTI/JCF doesn't mean "out of touch with feelings". The Thinking function is how we logically sort through information; the Feeling function is how we sort through what matters to people. Having a preference for sorting logical information can put you out of touch with other people at times, but it doesn't necessarily mean you can't process emotion.
Likewise, 4 doesn't mean "feelings" any more than, say, Type 2, which is known for being outwardly effusive and, at times, emotionally manipulative. I'd think it would be unusual to see a core 2 or 4 as a Thinker, simply because these types explore self, others, values, and meaning by nature (hence the Feeling realm). Having a 2 or 4 as a fix is in other territory altogether--everyone has their own way of relating to emotions, and that's got more to do with the mechanics of the core type than any particular heart- (or head-) fix.
I don't see why being 4-fixed Thinker, therefore, would be something particularly remarkable and therefore worth commenting on. However, because I do type as both an ENTP and a 4-fixer, I can at least try.
Like I said, I feel like my emotional responses have more to do with my core enneagram type than anything to do with MBTI or my tritype. I feel like the 4-fix perhaps adds a desire to reveal oneself and one's inner workings (in my own case), but this doesn't need to be my "emotional reality per se". In my case, it's very often my mind; it's somewhat detached in nature; and it seems to get me mistaken for a narcissist when I try. There's a tendency to dwell on my depressing past (which could be Si-inf related more than 4); a tendency to seek meaning and depth; a certain openness toward spirituality; and a tendency to feel "different" (not in a positive sense--in a painful sense, which is mitigated by feeling special or gifted ONLY SOMETIMES).
While I can have strong emotions, I don't particularly feel the need to identify with emotions or create a persona based on them. Yes, I can be moody, but I likely would regardless of my heart-fix. I'm just kind of irritating like that. If anything, the 4 amps up my reluctance to change or hide the moodiness, as well as heightening my expectations of being hated (so as I write this, some part of myself is saying, Fuck you world, I ain't self-censoring no matter how much you hate me for it.). I've also got an affinity for sad things and dramatic presentation.
For the record, "sad things" is the only thing I see that doesn't especially link up to ENTP in some way or another. I can add more detail if needed, that's just sort of a shot in the dark.
Thanks for this - good to have that clarification in this thread.
To answer your question of why it might be noteworthy... as I see INTPs and ENTPs writing about it, I wonder if it might be more noteworthy for Te than for Ti. As a Te-dominant personality (ESTJ), I am hyper-focused on the flow chart of what needs to get done and what steps need to be achieved in order to get there. This means that everything is prioritized and arranged for efficiency and effectiveness - including feelings and emotions (mine and those of others). That doesn't mean I'm cold and unfeeling or without compassion, but it does mean that I have a hard time catering to emotions for their own sake. There has to be a reason that makes sense within my cost-benefit framework. For example: I need to take care of myself emotionally right now, because if I don't, I'll cause damage (will feel worse, will make other people feel bad, will decrease my own productivity significantly).
To my knowledge, enneagram 4 doesn't like frameworks like that. 4 is an other, 4 operates outside of everyone's structures, because 4 is an exception to the general rule, as a rule. The idea of sacrificing one's emotions to a higher goal* - higher than their feelings, which they may define themselves on - is anathema.
My positive experience having a 4 fix is that I feel that I can have more compassion for others, and more realistic standards for their behavior, the more I understand my own inner world. I also give much better advice - and as an ESTJ, boy do I love giving advice.

Also, ESTJs are already the sorts of people who will be the lone voice standing against the unjust tide, and having a 4 fix gives me firmer grounding in my own values as I do that.
My negative experience having a 4 fix involves a whole lot of mental gymnastics - to the extent that I denied having a 4 fix at all, for a long time. Having an "inner self" that's more fragile, volatile, and dramatic than the identity that I prefer to operate with, and trying and failing to reason away the volatile/sensitive side as "just being sensitive" and not being based on anything deeper. What usually ends up happening is what I described earlier in this thread, namely being extremely moody but unable to accept that moodiness into my list of priorities - and having that moodiness leak out nonetheless. My friends have told me that they knew I had a connection to 4 the minute they first heard me aggressively (read: emotionally) deny that I had any feelings at all.
Also, I tend to rationalize 4-ish introspection as "if I figure out my inner world, it will be less chaotic and easier to predict" - leading to lengthy bouts of "productive" but ultimately pointless wallowing. (Most of why I have a new blog is because I got so sick of my old blog's pointless wallowing. Trying to keep myself from doing that too much.)
I figure cognitive dissonance is important to any enneagram type and type combination, but due to my own personal bias (i.e. being a TJ with a 4 fix), I find it particularly interesting.
*Enormous caveat here: I'm talking average to unhealthy 4, here, because I know that healthy 4s don't do this.