My perfect solution was to add two rings of 1/2 inch thick climbing rope, loosely wrapping the OTA above and below the altitude trunnions. The tiny bungee cord between is just keeping the lower ring from sliding and allows quick disconnect. It's not for carrying!
I can grab both ropes with one or two hands on any side of the OTA and reliably hold and carry it in any position from horizontal to vertical conveniently and securely.
As you gather and grab two ropes they are forming a self-adjusting triangle balancing the weight of the OTA perfectly regardless of the weight shift caused by attached equipment. Thus you can grab it even from a vertical position it will go to the horizontal smoothly. And it's easy to control singlehanded with just a push of a finger on the OTA from under the rope.
Grab the lower ring from both sides and you can carry or handle the OTA vertically with ease (e.g. I'm sometimes reversing it in the base to have the EP pointing downward). You can easily flip it upside down as well moving the grab from one rope to another (I'm storing my OTA upside down in the closet).
Disconnect the bungee cord and you have 2 independent handles in any place of the OTA to carry it with the friend over steep terrain or for better control (e.g. if you need to put it on a high shelf).
It turns out to be as well a perfect way to carry the scope at the observing site from location to location without disconnecting the OTA from the base! All I need is to put the OTA to the vertical position, grab both ropes with one hand on its front side and by the base handle underneath with another. Everything is naturally balancing as you lift it. Yes, it's heavy but totally doable if you need to move a bit in order to escape some tree branches interfering with the view.
Such handles works equally well on top of any soft scope cover which you may want to employ during the day break. In fact, they are helping me to better secure that cover in place too. Just tuck it under one or both ropes and it will withstand almost any wind.
Finally, I'm conveniently using these ropes for slow motion control of the scope in Azimuth near the Zenith, especially when pointing it with Telrad. Just pull one rope until it grabs in the needed direction (by the bungee cord and rope knot over the trunnion).
These handles are cheap and trivial to make, lightweight, easy to remove and install without any tools or OTA modifications.
UpdateSince that time I have upgraded my rings with the decent 12.5mm retro-reflective climbing static rope. Even more secure for grabbing and helps to see my setup from up to 150 feet away in the total darkness with just a dim red headlight. Important for my solo trips as it often takes several trips to the car and back to bring everything I need.
Ends of the rope are joined for about 4" of length by tightly wrapped around them thin nylon cord. That not only helps to avoid using bulky knots but also adding wider and stiffer handles, which adds even more versatility (e.g. more control, less strain when hauling for really long distance).