James Thomas, a bike fitter, occasionally collaborates with Francis Cade for YouTube videos. He recently described 9 bikes that he hates as a fitter. The first one was the Specialized Aethos, which he said "really ... annoys me ... this was a bike that was touted by Specialized as 'a bike for the people'. Why ... is it built like a ... race bike?" He goes on to say that it's got the SL7's geometry, which is much too long and too low for most consumers. Average riders would need a lot of spacers and a short stem, compromising the intended handling.
You are going from the Triban RC500, which I believe is an endurance bike, to the Bianchi Specialissima, a road race bike. Let us instead consider this three-way comparison between the Triban, a 57cm Specialissima, and a 57cm Infinito, their endurance bike.
The Specialissima would be 10mm longer and a whopping 47mm lower, and the 55cm is even lower than that. 47mm is already a huge change. Many bike brands specify no more than 40mm of spacers below the stem. So, it is not clear if you could keep your current position on the Specialissima. Actually, in general, people with shorter torsos relative to their legs are harder to fit, because the reach is the limiting factor and we don't want the stem too short. You would be using something like an 80mm stem, which is definitely on the short side.
If you in fact want to change to a performance road position, then I would recommend seeing a bike fitter. You could set yourself up for discomfort and possibly injuries if you change to a more racing-oriented position. If you're in your 20s and you know you want to road race, you can try this, and many people can adapt, but maybe try with a less expensive bike. It would also really be better to do a test ride on the bike first - as long a ride as you can arrange, not just a spin around the parking lot. Furthermore, you can ride very fast in an endurance road position.
The Infinito is likely to have slower steering (i.e. more trail, which means more stability) than the Specialissima. If you started riding on a performance road bike in your youth, then this may be something you dislike. However, most average riders are best served by an endurance road bike rather than a performance bike.
Glancing at the specs for the Specialissima and the Infinito, both with Ultegra Di2, it's not immediately clear why the former retails for US$3k more than the latter. The site says they have the same wheel spec, and the difference in the tires and the rest of the components doesn't seem that great. The layups might be very different (i.e. the layup schedule for the Infinito might be markedly less complex, which would add weight). But I don't know. I'm not familiar with Bianchi.
Yes, you would get the Specialissima name, which I assume is more prestigious than the Infinito. This is not nothing. But would it mean much to anyone apart from a serious Bianchi fan? Also on other subjective factors, the Infinito has Bianchi's more traditional Celeste rather than the darker green, and its name is much easier to type (although I believe only one syllable fewer in speech).
For reference, here's the comparison between the current bike and the 55cm versions of each bike. I lean towards recommending the 57cm Infinito, as it's the closest in stack and reach to the original bike.
