The post looks reasonably adequate as it was originally built even by modern standards. If it wasn't you'd have seen disaster long ago, when it first started to decay.
To maintain the style detail I'd rebuild to match, using pressure-treated lumber. By doing so you eliminate the need to cut that notch and you end up with a more robust post. A single post ...
The fire cracker idea sounds more fun than my idea, but I'd simply dig out the concrete and remove it whole. If you're lucky they only used 40lbs, that's not that bad. Once the concrete is unsupported by the soil, it should crack a bit easier for disposal.
The tool of choice here is a jackhammer. Today's electric ones will do the job nicely. You should be able to rent one and a portable generator to power it from a local tool rental shop. They should be able to fix you up with the correct tool for the job.
Strong winds or heavy snow will destroy that structure as you currently have it planned (if it would even be able to support it's own roof).
Imagine that this structure is not a structure, but a stool - if you sit on a stool where the legs are not tied together somewhere other than the top, the legs of the stool will want to push outward at the bottom, and ...
+1 option #4, plastic standoffs.
"Any thoughts on the most robust way to anchor the posts?"
You said you are renting and can not anchor.
Take some five gallon buckets and mix up some cement, fill the buckets with the cement and put an eye bolt in so it stick out of the top. Put a nut and washer on the threaded part of the eye bolt that will be in the ...