Sure, you can do that. The disadvantage of picking up the green (fused) circuit is that if the fuse blows (for whatever reason, including the fuse failed), your OD won't work. The OD manual talks about not using a fuse, due to the possibility of overspeeding the engine if the fuse fails.
Just as an example, my TR3 developed an intermittent short while on a 3-day run with the club a few years back. Since nothing critical is powered by that fuse (which I've replaced with a self-resetting circuit breaker), I was able to simply continue onwards with the club. We were on our way home before I finally figured out where the problem was. (Hint: I don't have the wiper motor installed.)
As long as you are considering modifications, I added an in-line fuse to the other wire to the relay. That will protect against shorts in the hot wire to the solenoid, which IMO is more likely to have problems than the hot wire to the relay. But it will also protect the solenoid if the cutout contacts fail to open for whatever reason. Solenoids are cheaper now, but they used to be upwards of $100.
I also added a diode across the solenoid, to help the relay contacts live longer. The original relay had oversize contacts I believe, to handle the inductive kickback from the solenoid; but none of the replacements I tried lasted more than a year, before adding the diode. Now, the same "cheap asian import" general-purpose relay has lasted more than 10 years.