Remember that matching is not necessarily about finding pairs; it's about paring down your data set into two comparable groups (if you agree with the position taken by Ho, Imai, King, & Stuart, 2007). So matching multiple controls to the same treatment is okay as long as you are correctly throwing out the controls that are incomparable to your treated units. You can also assess balance on your matched data set; if you have achieved strong balance, you don't need to worry much about bias (as long as you have achieved balance on the necessary variables).
Another approach you could try is weighting. Some control units will be up-weighted, and some will be down-weighted to form an equivalent group to your treated group. With your fairly large sample size, entropy balancing might be an effective way to proceed.
Finally, you can think about your estimand. The ATT is the difference between the treated units' outcomes and what they would have been had they received control. But you could also estimate the ATC (average treatment effect on the controls), the potential effect of being treated for those who didn't receive treatment, which is a viable but often overlooked estimand. If your eventual research question is, "Should I impose treatment on those who don't normally take it?", then the ATC is your estimand of interest. In that case, simply switch the labeling of your treatment and control group and proceed as normal, leaving your control group intact.