Not a pro but would be good to check the duct work to see what you have available to you -

there are lots of ceiling mounted hoods, even flush ones - not sure the efficacy but perhaps worth a look - or you could look at ceiling mount chimney style, it would hang down into the opening but perhaps work better

Personally I would be closing off the opening, put in a proper fan vented to the outside if you can and if not then go with a recirculating fan.

I’d agree with Lorraine to close up the opening and install a vent hood.

Yep, either close off the opening or move the stove. With the opening there you are essentially peeing into the wind with whatever you do. Nothing will work well.

The oening will need to either be closed or you get used to a proper hood hanging in the opening. Any ceiling mounted one will not be much good. Can you get into the ttic to see if what you have is actually venting to the outside?. We do need to see the whole kitchen please .

Start by checking the size of the current fan's duct. Bathroom fans often require only a 4" duct while range hoods require a minimum 6" or 3-1/4" x 10" duct.

The bottom of a range hood is generally 30" above the countertop/range so a range hood would extend into the opening to your living room. This would look awkward from the living room; I would consider reducing the height of the opening so the range hood was not visible from the living room. Then you could install a wall-mounted range hood.

You don't state the width of the space between the cabinets. Generally a hood fits between the cabinets (range width plus 3" on either side) but the space between your cabinets appears to be much wider.

A 46-inch gap would generally require a commercial type hood that in this case would have to be embedded into the ceiling and attic. For residential hoods, the window needs to be closed off at least 10 inches down, and the base of the residential hood set at the 36-inch level. Then, the usual stuff applies. [Extracted from another recent post I addressed.] Residential hoods assumed. En brief:

• The hood needs to overlap the cooking utensils by several inches due to plume expansion with plume rise. This applies to front to back depth as well as width. Hood base height above the cooktop between 30 and 36 inches is recommended. For higher spacing, larger hoods with more air flow rate are needed.
• The air flow rate into the hood's entry aperture should be of the order of 90 ft/min, which equates to 90 CFM per square foot of aperture.
• Assume that with baffles and make-up air (MUA) pressure losses, the actual flow of a blower will be about 2/3 of its rated flow.
• Duct diameter should be such that the flow velocity in the duct is between 1000 and 2000 ft/min at full power.
• If you want quiet, it is necessary to use an external or in-line blower with a silencer in the duct path between hood and blower. This suppresses blower blade tip turbulence noise and some downstream duct turbulence noise. Baffle hiss will remain.

This means that if your new hood is 2 ft x 3 ft at the entrance, the resulting 6 sq. ft. x 90 ft/min equals 540 CFM actual, requiring a blower rated (in free air) at 810 or thereabouts CFM. You would likely want an 8-inch duct in the attic. We are assuming an attic exists above the hood and not a bed.

MUA system details are another subject area to address after the Hood system details are preliminarily determined. MUA will need to be defined before ordering hood system parts due to possible interacting requirements. MUA needs to be introduced in a manner that minimizes additional turbulence in the cooking area.

Reviewing many of the hood related threads here will improve awareness of what is needed to keep grease off of walls and ceiling.