I would spring for the 200 amp panel. Less cluttered, more capacity for future upgrades, good for re-sale. I generally value improving infrastructure over saving money, within reason.
It is generally not the 200 AMP panel that puts it outside of reason. It is the new run of cable from the power meter to the panel, if not back to the transformer at the pole, and possible any grandfathered items that also have to be brought up to code.
Larger panels can be had without more larger service, sub-panels are also an option and 200 amp panels can have the main breaker reduced.
I too like to upgrade the infrastructure of the house, I try and make it every other project. But it has to make sense for my use. If the OP has lived in the house for a while and does not have issues with power consumption 200 amp service will not do them any benefits, unless they have plans that would increase power consumption.
To the OP.
If your current panel allows tandems then I would proceed with that route (some do, some only in certain slots, and some don't ... there is a reason they are called "cheater breakers"). If not you could replace the panel with a larger one, one that does accommodate tandem breakers, or you could add a sub-panel and route/reroute some circuits there. (I have a sub-panel in my garage workshop and most of the garage is on that panel, with the exception of the lights near the panel so I can have light when working on the panel).
Regarding the double tapped breakers, there are some brands that make breakers specifically meant to take 2 wires, if your panel is one of those brands, otherwise simply connected the two wire and the breaker by a pigtail is often a viable solution.
