If your intervals that you divide things up with do not overlap (for example, all response variable end up in disjoint bins, such as [0,2.5), [3.5,4.5), [4.5,5.5), etc), I would actually suggest you disregard the interval censored aspect of your data, and merely treat it as ordinal/discrete. And my bias is toward using interval censored methods!
The reason for this is that when using non-parametric and semi-parametric interval censored data estimators, if the intervals do not overlap, your results are exactly equivalent to the results if you had treated them as discrete ordered outcomes (ie 1 = [0,2.5), 2 = [2.5,3.5), etc). As such, special software is really unnecessary; you could easily use R's ordinal package or even coxme for mixed effects models.
If for some reason that doesn't currently make sense to me, your response intervals were not overlapping (ie for some reason you believe subject 1's exact time was in the interval [6-8), but you also believed subject 2's exact time was [7-9)) OR you're really committed to using fully parametric models, you can fit interval censored regression models (fully parametric AFT models can be found in the survival package, non-parametric, semi-parametric and fully parametric proportional odds and proportional hazards models can be found in my own icenReg package).
But I'm not aware of any mixed effects for models interval censoring data at the moment. If you really wanted a parametric mixed effects model, you could hand code your model into something like Stan or RJags (my understanding is that they both have syntax that allows for interval censoring). But I would strongly suggest using the ordinal or coxme packages.