Need Help Booking Luzern-Interlaken Express train with Swiss Travel Pass

If you have a Swiss Travel Pass, you do not need to book a ticket. You just get on board and show your pass when the conductor comes by. That is the way it works. If you want to buy a seat reservation, you can do that at Zentralbahn website.
https://www.zentralbahn.ch/en/experience/leisure/luzern-interlaken-express

Thanks, Sam! So if I’m concerned that any trains will book up, I just need to get a seat reservation in advance?

Swiss trains don't "book up". The Luzern-Interlaken Express is not just a tourist train. It is hourly regular service between the 2 cities and towns in between and locals ride it to get to their jobs or visit friends and family. Now you might have to stand, or not find seats together. That is why they sell seat reservations. They know the tourists will be worried and they can make a little extra money. July is busy with tourists, so it might be money well spent. If the train is connecting with the Golden Pass Express in Interlaken, it could be very full.

The only thing you need to do to make sure you are on a train is be at the station on time.

Trains are mass transit. You just find your train, get on, find as free seat (or stand for a bit in the rare case there are no free seats). Your pass allows you to use this nation wide mass transit system as much as you like. There is nothing special you need to do. There is no need to book trains, to check on, to negotiate gates or turnstiles. Just walk to a vehicle and board it...

The SBB app shows the consist (cars) of each train and an expected level of booking for each train. We did first class, never sat in a car that was half full either last summer or in January just past. Also never saw any appreciable crowding in the 2nd class cars. No need to reserve seats other than the "Scenic Express Trains" if you intend to do any of these. I believe you book (and pay) for those seats starting 90 days out. Think Bernini, Glacier or Golden Pass trains.

All of the trains on the Luzern-Interlaken Express are "scenic express trains", with carriages having diagonal viewing windows above.

You don't say which direction you want to travel. Going from Luzern, trains at 7:06, 9:06, 12:06, and 14:06 connect to the Golden Pass Express to Montreux, so they are likely to be more crowded. Going from Interlaken, trains at 11:04, 13:04, 16:04, and 18:04 connect from the Golden Pass Express, so they are also likely to be more crowded.

Last January I rode from Interlaken to Luzern on a 1st class pass and I was the only one in the carriage. There was one young lady who came in and sat down, but the conductor kicked her out when he saw her 2nd class ticket.

In July, getting a seat reservation in Standard (2nd class) carriages is a good idea as it’s peak travel season and the route is popular. I have never seen First-class carriages even get half-full.
Before understanding the seat reservation system, I was once on the Glacier Express route without any seat reservations ( the train left Zermatt half-empty) and kept needing to move for passengers who had reserved seats and joined the journey at intermediate stops. There were seats available throughout the entire journey.

In Summer the trains on Luzern - Interlaken are lengthened, and are then 10 cars long. Most of those cars are non reserved, you cannot even reserve seats in them.
That often leads to the odd situation where tourist with reservations end up sitting in a more crowded car then people without reservations.

My suggestion: Don't bother with reservations. They are overpriced anyway. Just head to the front of the train at Luzern, where there are normally always free seats.

There was one young lady who came in and sat down, but the conductor
kicked her out when he saw her 2nd class ticket.

The railways have stopped doing that. Now if you are caught with a 2nd class ticket in 1st class you have to pay the upgrade.

Thanks to everyone who has given me such good advice! I really appreciate all of you!!