The problem with most dog trailers is they have a limited weight capacity and often do poorly on rough roads.
You might get the Burley Travoy to work for you:
This Travoy cargo bike trailer is ready for use on your everyday commute or the occasional bikeshare ride for any urban adventure.
burley.com
If your rear axle is normal or semi-normal you might look at one of the Burley trailers use their hitch system, which is, um burly:
The Coho XC single-wheel bike cargo trailer brings durability and thoughtful design to gear-hauling for bike camping, touring, and singletrack riding.
burley.com
Note that the Burley trailers are expensive. On the other hand you can probably then get away with regular camping gear which you might have or can borrow.
For bikepacking and bicycle travel in general it is more about minimizing volume than minimizing weight. Bikepacking bags are usually quite small compared to backpacks and are awkwardly shaped. So you literally need to Tetris in your stuff.
You
might be able to zip tie
3-pack cage mounts to the front fork (
very carefully!) and the rear stays, and mount 5L dry bags and get 20L of storage. Then add a
handlebar roll and you can plausibly make the trip.
On the route (or routes) you are describing you should be able to comfortably resupply every day at a supermarket. That lets you eat better and means you never have to carry very much food, or likely carry it very far. Since you'll need to recharge that bike pretty frequently chances are that won't be super inconvenient to your plans.
Let us know what you come up with. Yeah, a lot of this crap is expensive.