I know this is a super old post but I was having the same issue as the OP'S not being able to see my computer at night/early morning.

I've been using a Lezyne super GPS which although is definitely much more than what the OP was looking for, allow me to turn the back light on and leave it on as long as I like without requiring period button presses to keep the back light on.

Also I had a sigma wireless bc 16.12 with cadence that also has a backlight but it's backlight will turn off after a few seconds. Great computer but I wanted an upgrade.