When I make a bow, I shoot it a dozen or so times between scrape and check sessions once it is one the short string. I do this at my waist instead of proper shooting form and mark the arrow so I don't draw above the target weight. I feel that all you need to register a change in tillering is the short snap of an arrow release.
In your case you could put the thumb of your bad arm through a belt loop to hold the string and push the bow out with your good arm. Sounds nutty but I bet it would work.
I think an MRI is in order, what you described is exactly like my surgeon described when he looked at my neck MRI a few weeks ago. He said saw an area that needed work and asked me if I was feeling symptoms like you described, fortunately I wasn't.
My latest set back had me thinking my fake hip was failing big time, terrible pain right in the ball of joint, at times I couldn't walk 20 ft with out that lightning bolt hitting me in the fake hip.
An MRI dispelled my theory of having my fake hip failing, the pain was generated by two bulging disc at L4 and 5 but didn't manifest itself until it got to my hip region. I got the shots in my problem area 3 weeks ago, it didn't appear that they worked but a week ago I felt something shift in my back and the pain started fading away, so far so good, I walked 2 miles of very hilly ground yesterday with no pain.