One option you cited is bagging.
A second option is to cut the grass and simply discharge it onto the lawn, allowing it to dry for a while in the warm sun. A bit later, pop in the mulch plug and mulch away.

A variation of the second option would be to try making a "first pass" cut with the blades raised very high, so you are only taking a bit off the top of the grass.
Mow the entire lawn at that height and then drop the deck to the desired finish cut height and mow the second pass.
You can also experiment with trying a slower ground speed so that the deck is dealing with a "smaller feed volume" of grass in the same elapsed time. This is analogous to the strategy one must use to handle grass which has gotten too long because a scheduled "mowing date" was missed.

Of course the second option effectively doubles your labour (and petrol consumption) but it does permit you to reap the benefits of mulching natural nutrients back into the lawn, and not having to discard (or compost pile) the clippings.

Other forum members may offer some other advice (perhaps even recommendations for specific types of specialty mulching blades).

You'll have to do the math to decide what's best for you Andy.