How best to hang these planters on fence panel

If you want them both at the same height and beside each other [similar to mine] then I'd use something a little bit heftier. The timber in your pic is what I use for the battens on the pergola, and on my other screen.
I have a horizontal piece of 4 inch timber for mine to sit on. You can just see it in the 2nd pic, in between the two baskets. It's screwed onto the two posts as the trellis wouldn't really have been sturdy enough, because although it was very sturdy stuff, it was recycled from the existing fence, and was going to be supporting a fair weight.
If you don't need them to be in line, or at the same heights, you could certainly use that timber, but you'd need to be sure it's properly attached to the existing fence, and you'll just have to measure up to see that the hooks will fit in nicely, because only you can see how the sizes will work. If not, then the short vertical batten at each end of the basket [attached to the uprights of the fence] with a horizontal on those would be better.

 

Thanks @fairygirl helpful as always. I can see the horizontal piece in your second piece now that you mention where to look, my eyesight isn’t that good so I didn’t spot it initially.
I’m thinking of one planter basket (we have two) only on each fence panel so I may get away with something not too hefty.
But I think I have a much better idea of what needs to be done.
Thank you

 

Sorry @NoviceGardener2023 - I should have mentioned the other piece of timber! I knew the troughs would be heavy once filled, and I would never have trusted the trellis to withstand the weight, especially when they were recycled from the original site and were a bit worn out in places.
If it's just one being hung on each panel, you should be fine with that timber you intended getting. I use it for various other things around the garden as well as the pergola and the other screen, which has been in situ for over a decade and has climbers growing on it.
I don't think your troughs are nearly as big a volume as mine either, due to the general shape, and therefore won't be so heavy. If you wanted a bit of extra 'heft' you could add another vertical in the middle of the support, if you have to go for that option.
Good luck with it though