A simple two-primer dimer can't be much longer than the two primers stacked end to end, so 40-50 bp for typical 20-mers. You can't get to 190 bp that way unless you're forming concatemers, which does happen with plain Taq and a lot of cycles, but it wouldn't be my first guess.

The observation I'd actually lean on is that your NTC band migrates differently from the sample band. If reagent contamination were the problem you'd expect the same product at the same size. A different size means a different molecule, which points at primer-driven amplification rather than stray template. The Ta sensitivity is consistent with that, though on its own it's weak evidence, since contamination amplicons shift with Ta as well.

Cheapest way to stop guessing is to gel-extract the NTC band and Sanger it. Ten dollars or so and you know definitively whether it's your target (contamination) or something else (artefact). Considerably less than a redesign.

On suppressing it without redesigning: you're on standard Taq, which is active while you pipette and during the ramp, and that's exactly when dimers form. Moving to a hot-start Taq is a reagent swap rather than a redesign and it's the biggest single lever you have here. After that, lower primer concentration and cut cycle number.

Before spending anything though, check whether your two primers have 3' end complementarity, since that's what drives template-free product and it costs nothing to look. I work on SeqBench so weight this accordingly, but https://seqbench.com/tools/oligo-analyzer will give you the cross-dimer and hairpin alignments and flags the structures that involve the 3' end, which are the ones that matter. Any oligo analyser does the same thing.

For your qPCR question the usual bar is NTC Cq at least 5 cycles behind your lowest sample, with a melt peak clearly distinct from the target. Within 3 cycles and I wouldn't trust the assay at low input.