Before you go where you don't need to go...
Where do you intend to plug in the I/O Linc?
Is it possible to run a 4-wire cable from the current switch to the I/O Linc?
That's actually a good solution. Add one surface mount box, an easy throw switch and you even have room to add more batteries in parallel so power will last longer.
Great find and great idea.
You want something like this:
https://www.itead.cc/inching-self-locking-wifi-wireless-switch.html
And if course you will need to add your own battery and a way to monitor power.
I suggest you set on time to a time you will be able to watch the program and test it. Maybe you can see something in the Admin Console that will show you why the lights don't stay on.
I have created two test programs and I'm waiting to see if they both trigger. It will be about 5 minutes to turn on time.
Test 1 just turned on the light. Wait 10 minutes to see if Test 2 turns off the light.
The lights turned on and off.
I have no idea why your programs are not working.
Did you notice that in your screenshot the Lights Off is running. I suggest that the Off prgram is turning the lights off as soon as they are turned on.
There are some that have the power of ISY. There are some that have have the customer support of ISY. One would have to research to know where each hub stands. For instance HomeSeer may be a better fit for some.