Water based adhesives have the following disadvantages:
Slower drying than solvent based adhesives
Susceptible to freezing at low temperatures.
Lower water resistance than solvent-based products.
On the other hand, disadvantages of solvent based adhesives:
Cost – solvent based adhesives are more expensive than water-based adhesives.
Cleaning: Residues ...
I use water-based (project or panel) adhesive on foam board or any plastic which the solvent will dissolve, or where cleanup of spills is a concern. Solvent-based (heavy duty) for almost everything else. It dries faster and holds better.
The hot melt glue might work but I think there's an easier alternative. I would try clear silicone adhesive / sealant caulk for this, it's cheap and you know it's weather proof.
I have extensive unintentional experience with silicone adhesive and fabric. I have noticed that when I get a spec of silicone on my clothes, it's pretty much permanent, you can ...
Consider using a double-sided clear mounting tape. 3M makes one with a clear but rubbery and flexible carrier that makes an extremely strong bond. Since it sticks immediately, you don't have to spend time holding it. Remember that tapes of this type have a bond strength that improves greatly over the first 24 hours, so if you do a small area as a test, give ...
Strengthen the supplied sticky backing with something better at all stress points (beginning, end, change of direction, near any switches or other places the strip will be handled).
double sided foam tape
A bracket made of a strip of wood with wood glue and/or screws
A metal bracket ... just a thin strip of metal that runs across the light strip ...
I don’t think glue is good for framing in vegetable gardens. The glue could be toxic and bleed into the food.
I’d be careful of the type of pressure treated wood to use too. Some types are no longer allowed and could contaminate the food.
You're not building a deep space probe where you need to optimize everything as much as possible.
Screws and nails are plenty suited to the job. We've been building buildings with screws and nails for thousand of years.
If the wood is rotting, solve that problem. It won't matter what means of joining you use when everything is rotting anyway. Use good ...