...... I recommend to turn off the Hume before Holbrook and head up through the Snowys and come into the back of Canberra very nice ride
We're in the ute and towing a caravan.
Helen & I took Jeep Grand Cherokee and the caravan on the road from Holebrook to Cooma early January. The road surface was overall okay (a few pot holes, some road works) but was windy and very, very steep and hilly, especially the section of the Elliot Way in the link.
https://maps.app.goo.gl/QTGt6HREQDzriaEe6Even with the Jeep's torquey diesel V6 towing the 2.5T (loaded) caravan, climbing some of the steep sections required 2nd gear through some of the hairpins, and on some downhill sections, 2nd gear engine braking was also required. After Kiandra the Snowy Mountains Highway was a big improvement.
Jeep and caravan aren't a small combo.

If your combo is not much bigger, you should be okay; if smaller no problems.
If you decide that route, the caravan park at Tumbarrumba is very nice (cheapest powered site on our holiday, very good amenities) and a short walk to the shopping centre (pub and bakery).
Adaminably is a good lunch stop. I made a mistake by getting a pie (dry and stale) at the bakery, although the coffee was okay. Helen's choice of a burger at the Small Fry Takeway, was spot on, she reckon it was one of the best burgers ever.
https://maps.app.goo.gl/KkR9HZm8EaFsHbzr5