Wiring – Three way switch with common terminal hard-wired to traveller

Can I install the timer in place of this switch and tie the load side of the timer with both travelers leaving the box?

Yes (probably) and no. Unless the timer pulls an amount of current that would put the circuit over 80% of the 15 A supplied by the breaker (12 A) you can install the timer on the original circuit but don't tap the travelers together.

You can't run 14 AWG wire in parallel, which would be the case if you tapped them together. This is done when dealing with larger wires where the cost factor compared to additional ampacity isn't economical.

NEC 310.10(H) Conductors in Parrallel

(1) General

Aluminum, copper-clad aluminum, or copper conductors, for each phase, polarity, neutral, or grounded circuit shall be permitted to be connected in parallel (electrically joined at both ends) only in sized 1/0 AWG and larger where installed in accordance with 310.10(H)(2) though (H)(6).

Note: You can actually run conductors smaller than that in parallel. Those exceptions don't seem to apply here.

Is an alternative to cap the red traveler, install the timer in place of the first line-side three-way, and use the black wires throughout the circuit? This would leave the second load-side three way as a dummy?

Yes, this is the correct thing to do based on the above information. However, I wouldn't leave a dummy switch there. Just remove the switch and replace its cover with a blank plate.

Is there another option?

Well, there's a couple additional options depending on current and future plans for operation.

1. If the device is capable and there's a need, you can use the red as a second timed leg. This would give you the ability to use it for some future fixtures. Obviously, additional work would be done.
2. You could use the black for a constant hot and the red for the switch leg and use the black to feed a single-pole switch as sort of an override. The timer would turn them on whether the switch is on or not, and the switch could turn them on without manually switching the timer circuit any time you need.

It sounds like the voltage tester has issues. There are other ways to identify the wires.

Usually 3/4-way complexes are a mess (not least because the wire color does not correspond to the purpose in any way). But your 4-way switch makes this surprisingly simple: both pairs of messengers are on the red+black wires.

So go to the remaining 3-ways. Find the red+black pair that goes into the same cable: those are the messengers. The remaining wire is the always-hot or switched-hot.

You will also note the messengers are on brass screws, and the always-hot or switched-hot is on the black screw.