It was VI that had its budget reduced (at least initially. Think it might've eventually had the same budget as V) . V had an increase from IVs budget (as IV had from III, and III from II) I think Shatner wanted more for the climax of the movie but it wasn't granted.
Yeah, I think from my understanding of it V was budgeted higher than IV. It's just that Shatner's original concept overestimated the money that Paramount was willing to let him play with, so he had to start trimming the fat. Which meant bye-bye Rockmen.

VI had the same budget as V, but they were just cleverer about how they spent the money. It was an incredibly tight production.