There is no absolute reason that an LD could not be found at this age. From your description, he appears to demonstrate significant underperformance already. The usual reason we wait until later is because of the antiquated "wait to fail" structure/philosophy of public schools (RTI is an attempt to circumvent that, by remediating first, and diagnosing later).
OTOH, re-testing at this point would create some challenges mainly with cognitive instruments, depending on what is available to the school system. If he was tested on a preschool instrument, then the school-age instruments are still available. (Or the WISC-V, if they've bought one. Mine arrived over a week ago, so I know they're shipping pre-orders now.) Achievement can be re-assessed with the same instrument after six months.
A second issue (probably more important) is that in the absence of data suggesting that the previous testing is invalid, best practice would advise against re-testing, most certainly the cognitive, and probably the academic achievement as well.
I understand your concerns as a parent, but from the perspective of the school, he has only had about 4 months of schooling (and a month of it was rather sketchy end-of-year/beginning-of-year school) since his previous assessment. It is difficult to predict that enough development will have happened to make his learning profile clear enough for effective accommodations to be developed. Was your testing private, or did the school participate in that? Did any concrete recommendations come out of it?
