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Here are the two best connections for your flight. Later during the day they are a little easier. From Bürgenstock you don't have to go all the way back to Lucerne in order to get a train to Engelberg, the concierge will find you a quicker trip.

The snow should be no problem as there is more than plenty at present... I do hope you are a good skier, the start at the Titlis is not for beginners:

Engelberg 18.04.05 dep: 05:25 BUS
Wolfenschiessen arr: 05:41 switch to train

Wolfenschiessen dep:05:45, S4
Luzern arr: 06:16 on track 14 switch to train

Luzern dep: 06:21 on track 5
Zürich HB arr: 07:19 switch to train

Zürich HB dep: 07:26 on track 52 S2
Zürich Flughafen arr: 07:38

Engelberg 18.04.05 dep: 06:10 BUS
Wolfenschiessen arr: 06:26 switch to train

Wolfenschiessen dep:06:32, S4
Hergiswil an 06:51 switch to train

Hergiswil dep: 06:54 on track 1,
Luzern arr: 07:04 switch to train

Luzern dep: 07:10 on track 7, direct train to
Zürich Flughafen arr: 08:13

If you are not flying with an American carrier directly from ZRH, you might consider this connection which is much more convenient, especially if you have luggage. You can also consider checking in online and you'll make the connection easily. Trains are reliable

Engelberg 18.04.05 dep: 06:45
Luzern arr: 07:49 on track 14

Luzern dep: 08:10 on track 7 direct train to Zürich Airport
Zürich Flughafen arr: 09:13

You also have the possibility of checking in directly in Engelberg

http://mct.sbb.ch/mct/en/engelberg.pdf

PM me if you have more questions