No, you absolutely should do a starter with washed yeast. It may stay in the fridge a long time, dormant, so a starter wakes it up and ensures viability.
I wouldn't put that starter in the fridge tonight- I would pitch the whole thing at brewing tomorrow. Since it's at high krausen now, it'll not have a chance to settle out before tomorrow. If you stick it in the fridge now, you'll just encourage it to go dormant but it probably won't flocculate yet, and you'd have to pitch the whole thing tomorrow anyway. Plus, cooling it, then warming it up to pitch, would probably be detrimental as well.