In theory, yes you should be able to do that. However, I won't promise that it will work. Amtrak eliminated checked bags on all Northeast Corridor trains between Washington DC and Boston, except for one. That's the overnight run, train #66/#67.
So assuming that you get a redcap employee in Meridian who knows what he's doing, he should be able to check your bags to DC and then from there on the overnight train to Providence.
However, since your tickets for the Crescent only state a destination of Washington and Acela Express does not have checked bags, you may find an employee who either refuses to do what I've suggested or doesn't know that he/she can do what I'm suggesting.
Note: This idea only works if you keep the plan of staying overnight in DC. If you were to go straight to Providence without an overnight, you would arrive in PVD long before your luggage. In fact you'd have to come back to the station the next day for your luggage.