We'll be arriving (without our luggage, as we'll be using Express Baggage from Zermatt) at the train station in Lauterbrunnen at 13:00 on a Sunday in late August.
If the weather is good, will there be enough time to head to Splithorn from there, and if so, is this the best route?
In Lauterbrunnen you take the bus to 'Stechelberg Schilthornbahn' (about 20 minutes). Board the cable car here. You have to switch to another cable car in Gimmelwald before you reach Mirren. The final leg of the trip starts in Mürren. You'll board the cable car and switch to another one in Birg. Then you reach the summit of the Schilthorn.