How to jump from management consulting to product management role? Is it a good idea?

It depends on your motivation. However, it's better to switch within consulting or transition to a business/strategy role.

Too much supply of PMs would make the transition unfavorable. You don't have hands-on experience, the only way is if past engagements are relevant to the company's domain.
Think if you were the hiring manager, would you want to take a risk with someone inexperienced (in PM) at senior level?

Don’t do it. Consulting is end game for Product Management