You could make one classifier for each primer pair. At the very least, you should make separate classifiers for ITS and 16S, but you could also use a full-length 16S classifier (e.g., one of the pre-trained classifiers on the QIIME 2 website) to classify all of the 16S domains simultaneously.
Judging from the Qiagen website, it appears that they split the different primer pairs into separate datasets and classify these separately.
So I recommend you do the same. If primers are present in your reads, you can use q2-cutadapt to split up your sequences into groups prior to denoising. If not, denoise everything together and use qiime quality-control exclude-seqs to filter out individual amplicon sites by aligning against reference sequences trimmed to the same amplicon region.
I hope that helps!