Running AC/Heater when washing a cleanable filter
It is not advisable to run the a/c without a filter in place nor to run the washable one while it is wet. Damp may be OK but not wet.

The best practice would be to get another of the washable filters and rotate them. Alternatively get a disposable filter that you use while the washable filter is drying.

You could speed up the drying process using a shop vac to pull water out of the filter, but then you would have water in the shop vac and you have to dry it out. Don't put a shop vac away wet; it will grow mold.

Alternatively you could dry out the filter by first shaking it gently, then pressing it moderately between two towels or terry cloth bath mats, then going over it with a hair dryer, but be careful not to overheat it.