IMHO, your WAW should pursue you if things are to get better. Until that time, you should not make matters worse by asking her out on a date. It will show her a desperate man who is pining away for her. That is not what she should be seeing in you!
I know you miss her badly. But you won't get her back by pursuing her while she pursuing a D from the M. It took Crimson over a year after the D was final before his XW would even discuss being on a "date". But the last time I checked, they were trying to work things out. So, it may take the D and a year later before you can go there.
I maintain that a woman has ways to let a man know that she's interested in being alone with him. Until that time, he would do well to leave it alone. And.....if she is "showing" him she wants to be with him and he's too blind to see, then he doesn't need to be alone with her anyway.
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My question is, If she does contact me, is it appropriate at that time to invite her to something I may be involved in such as church, or maybe to go for a walk or what not. Is that considered asking on a date?
Yes, it's a very cheap date. But Crimson's XW had a problem even with the word "date".
Why is she contacting you? Is it always about business or kids? Does she ever contact you just to talk? Does she ever act interested in how you're spending your time or how you're feeling?
The LRT really works best, IMHO, when the couple is still under the same roof. However, you could go dark. That may work better for you. If you can't go dark (the really real last resort), then try to "drop the rope", which is the next best thing.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!
But Crimson's XW had a problem even with the word "date". 