You have several months to get used to the GSA, practice.
There are plenty of places in the San Juans that you could take your GSA with little problems. From Silverton it is an easy gravel road ride to Animas Forks, from there you will be able to gage if you want to push farther. The ride up to Ophir Pass is fairly easy, down the west side the first 1/2 mile will be rather hairy on a big bike (to say the least).
For me the downhills or much worse on a big bike.
There are more good paved roads in western Colorado than you can ride in a week so go for it,
Durango-Silverton-Ouray-Ridgeway-Telluride-Naturita-Gateway-Grand Junction is an awesome ride. Not quite the same as some of the jeep roads but you can do the easier jeep roads and get a taste.
As to your original question, I would say do not attempt riding to the top of Engineer Pass on a GSA until you have much more experience riding in gnarly places on the GSA. For sure do not attempt the Engineer Pass road that takes off of 550 just south of Ouray.
F800GS Orange and Black
KLR 650 (not new)