Lucerne 3 full days

You have Mt Pilatus listed twice, on Day 2 and Day 4. Or perhaps you meant Mt Rigi for one of them.

Edited my post. It is MtTitlis

Day 3 requires over 6 hours of travel time alone (in addition to enjoying time on the mountains). Take a look at SBB.ch

• Luzern to Stoos about 2 hours
• Stoos to Stanserhorn about 3 hours
• Stanserhorn back to Luzern 1 hour

Keep in mind the time of year that you are going - the Stoos ridge hike doesn’t open until later in the season, and that is usually the big drawing card for visitors to go there. Perhaps choose something closer to Luzern if you want to do a second destination on this day.

can't quite tell - "guest half of May" could be first half or last half - not sure, but it will make a difference in May.

a really easy to do day which is great fun for kiddoes and grown ups is to take the boat on the seriously beautiful Vierwaldstättersee (Lake Lucerne) to Vitznau. Then the Rigi Bahn cogwheel train up to Rigi Kulm, the summit, and then down the other side by a different cogwheel train to Goldau, then train back to Luzern. Or you could use the cable car between Weggis and Rigi Kaltbad.

Half fare card valid on all parts of the trip.

Good fun. Might even see a steam cogwheel when the season gets going.

https://www.rigi.ch/en/experience/mountain-railways

If you’re planning to do the Golden Round Trip for Mount Pilatus, that takes up the better part of a day. The Transportation Museum takes a half day. The exhibits are fascinating even for a casual visitor. Consider going to the museum on the day you arrive instead.

And keep in mind when check in is for your hotel. You may arrive before noon but your room will likely not be ready until 3pm. Hotels will let you check your bags, but then send you on your way until your room is ready.

Consider a cruise on the lake as an option if the weather doesn't cooperate..

Will be purchasing a Swiss Half fare card

Assume you mean two Half Fare Cards to cover the two adults and the free Family Card to cover the children,

Titlis with Stanserhorn on the way back to Luzern? Take some time at lake Trubsee while visiting Titlis. Another option is Stanserhorn on the way to Grindelwald.

The kids will have a lot of activities at Pilatus and could easily take up most of the day when considering the different options to reach the mountain top.