The main benefit of ductless systems is that they are effectively zoned so you can have different temperatures in different rooms. However, many contractors oversize them but then they are lousy at dehumidification.
If your main question is cost, we cannot discuss prices. Our costs for equipment has increased steadily and significantly since the pandemic. HVAC is a skilled trade so you do not want a handyman to install an air conditioner. A poor installation will cost more in poor performance and repairs than you might save initially.
As for geothermal, I just don’t think it’s worth the enormous installation cost. You may want to consider a modulating heat pump if you want to reduce your use of gas.

 

No one makes a unit small enough for that small of a space. A Ducted mini split may be doable, but we need eyes on to know.

The installing contractor makes or breaks the unit. Choose them wisely.

 

New England? So the furnace is in a cellar now? Yes?

Warm air rises. It rises because it is lighter. Cold air falls. It falls because it is heavier.
The warmest air in your home will always be at the ceiling level of the second floor.

If the furnace is in the basement now - are there full sized metal Return air ducts connecting the furnace to ceiling level of the second floor? To cool the second floor will require the equivalent of a metal duct about 10" by 16". Round metal would be about 14-15". Plastic flex duct would require about 18" inside diameter.

Is something like that now present? And if not; did the quoting contractor propose including something like that? If not; you have the wrong contractor.

Are you willing to give up that much closet space or build a corner or mid-wall chase that size to enclose the return duct?

Would a partial solution suit your purpose? Maybe a single unit located in the attic to centrally cool only the second floor?

Is this a 'cape cod' style house? Knee walls in the upper rooms? If so; you may want to go with a Unico system with it's 2" air ducts.


PHM
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Put in 2 systems, one for main, one for upstairs. It is often hard to put one system in to handle both floors with existing duct that was not designed for cooling.

It will also give better control of temperature on each floor and has the added benefit that if/when one system stops working you still have part of the house conditioned which will help with the down section.

This is not a simply project and will need someone that can do a load calc, design duct systems, then build and install. It may need to be done for both floors as the existing duct may have enough issues with size and design that it will be better to replace than reuse.

 

Your top priority seems to be cooling the bedroom, so I will consider all the options from that perspective.

As said above, adding an AC to your existing duct system - even with some duct upgrades - is not likely to ever cool your second floor well, so I would eliminate this option. It could work if you wanted to add cooling to your first floor only, but that doesn't seem to be your main goal for this project.

You could do a dedicated central ducted system for the second floor. However, the smallest one available will probably be oversized for a single floor, if your house is 1600 square feet. If the ductwork for this system goes in the attic, that will lead to poor energy performance. And cost would be high as new ducts would be be needed.

Finally, the last option is a mini split for the bedroom. Personally this is the one I would lean towards, given everything you've said. Yes, it would probably be oversized - even the smallest mini splits are usually oversized for a single room. However, mini splits are pretty much always modulating, whereas most ducted systems usually aren't, so the mini split would probability handle an oversized application better. At the very least, it would be a huge upgrade over a window unit which is probably also oversized. The mini split will be the most energy efficient option - using less energy than a ducted system or window unit. Finally, cost seems to be a concern for you and the mini split would be far cheaper than the central options.