Two partial days based in Interlaken

Less is more.
I like your 2nd plan for Day 1 when you arrive in Interlaken. On Day 2, why not just skip Schilthorn? Save the time and money by not trying to fit this in. Instead, soak in the scenery by taking the North Face trail from Murren which will give you views of the Eiger, Monch and Jungfrau. And then Lauterbrunnen Valley walk (or reverse, depending on weather.)

Is there a reason you need to stay the night in Interlaken or can you continue on to Lauterbrunnen and overnight there?

If you do that you can take the gondola up to Gruetschalp, cog train across to Muerren, then catch the Schilthornbahn up to the Schilthorn unless it's socked in.

If you get to the valley and the sky is clear, go up Schilthorn right away. Don’t assume day 2 will be like day 1. It could be cloudy, foggy, rainy your second day. It’s unpredictable, so take advantage of any good weather you can get.

Following on from Bets' reply, never assume about the weather, if the weather is good do all the stuff that is weather dependent as soon as possible. Don't wait an hour. Grab it.

The valley will wait. Interlaken will always be there. If you want to go up in the mountains, go when you can.

Do a revised version of the itinerary you linked if it feels a bit too much. I’d do this on the day where you can start out early as views up at Schilthorn are typically best in the morning. So leave Interlaken as early in the morning as you can to do this and you can be done and back to Interlaken to take a late train to Zermatt.

Train Interlaken to Lauterbrunnen > Cable car up to Grutschalp > Get on the train that goes to Murren but get off at Winteregg stop and walk from there to Murren (the most scenic section of that walk) or if you’re short on time just take the train the whole way to Murren, you’ll see the same views from the train (sit on left side) > walk through Murren to Schilthornbahn > Up to Schilthorn > Stop at Birg on the way down > Continue down to Murren > Option to walk down to Gimmelwald on the paved path or continue on cable car down if you feel short on time > Explore Gimmelwald for an hour > Cable car down to Stechelberg > take the bus through The Valley (optional stop at Trummelbach Falls if there is time and you want to wait in the line, because there will be one) > Continue on bus to Lauterbrunnen and then train back to Interlaken.

I don’t recommend hiking Lauterbrunnen Valley from Stechelberg to Lauterbrunnen as you’re walking away from the views.

For your shorter arrival day, drop off your luggage and do a boat tour on Lake Thun or Lake Brienz for the afternoon.

If the day you have allotted to Murren/Schilthorn is cloudy, skip Schilthorn and maybe do this….

Walk from Lauterbrunnen to Stechelberg (option to stop at Trummelbach Falls), cable car Stechelberg to Gimmelwald to explore for a bit, cable car up to Murren, walk through Murren and then train to Grutschalp, cable car down to Lauterbrunnen and return to Interlaken.

Appreciate all the thoughtful responses!

After reading through the advice, we switched up our lodging from Interlaken to Wengen, so we'll now be a lot closer to the action during our limited time in the area. I think with more central lodging and thinking about Schilthorn as something to do only if the weather lines up perfectly, we'll be in good shape.

Thanks everyone!