Water – Low water pressure at reverse osmosis tap after running fridge dispenser

I don't see any problem with installing it horizontally on the run between the two sinks. You'll probably need a new slip nut (the part that goes on top of the trap) but otherwise it can probably connect into the existing union that's there now. Just keep the piece that goes to the RO on the top (facing up).

You likely have air trapped somewhere it shouldn't.

Water doesn't compress so when you turn on the tap, the flow is the pressure from the city divided by the resistance of all the pipe between it and the tap.

If you have an air bubble, then when the tap is shut off the air will compress until the pressure matches that from the city. When you turn the tap on, the flow will be the pressure from the air bubble divided by all the pipe between it and the tap. Since there is much less pipe in this case, the flow is much higher.

The pressure in the air bubble quickly drops as the water is pushed out until you again match the original case.

This is especially noticeable if you have have a smaller pipe somewhere between the city and the air bubble than between the air bubble and the tap since obviously the flow is limited by the smallest (cross-sectional-area) pipe.

Depending on the size/shape of the filter, it could be air in the filter itself. See if you can fully fill it with water before connecting it.

If you shut off the house water while making the change, then air could have gotten into any branch going to a higher location than your sink. If you haven't tried running fresh water in every single location in your home, that may be all it takes to clear out the air.