Another regular Mull visitor here.
The English May half term is usually a good time, weather wise but also very, very, very busy.
Contact accommodation direct about short breaks, as most try to avoid Airbnb and such platforms. We've stayed at, or looked at, Treshnish farm, Cille Mor Lodges, Port nan Gael campsite (they have pods), and Saorphinn Cottages. Hotels will also do short breaks but aren't as easy with kids.
Tobermory Bay Tours are great for younger kids if you think a full day boat trip to Staffa and the Treshnish Isles would be too much (although the guaranteed puffins, and possible dolphins might be worth it!). You can take small passenger ferries to Iona and Ulva.
Calgary beach has an art nature trail and ice cream shop.
Tobermory is a good base with lots of places to eat out.
Isle of Mull cheese has a lovely cafe and farm tours.
Mull Pony Trekking takes wee ones on Shetland ponies.
Mull Museum and Duart Castle are good on wet days but really just bring your waterproofs!
Either book the Oban to Craignure ferry now, or consider going via the Corran Ferry (south of Fort William), then Lochaline to Fishnish or Kilchoan to Tobermory - all are turn up and go.