Couple of points:
Woven cloth is not paper, it's what K&N typically use. It's oiled cotton. Most OEM filters I believe are paper and UNI and a few others use oiled foam.
BS series carbs can work fine with good pod filters, but that said, most cheap pod filters are not what anyone would describe as "good" many have poor airflow and poor filtration and some partially obscure the air inlets in the carb and some make it impossible for the slides to lift at all.
That's why it is usually better to get a bike running well on a stock airbox with a clean OEM filter inside. It eliminates one variable. People will tell you that a CV carb cannot be jetted with pods or that they don't work on two strokes. Those are vague generalities that are not always true but are a good indication that maybe don't go down that road just yet.
120-125psi of compression hot, isn't spectacular but the bike should run well enough at those numbers. Old bikes never came with very high compression, but it's always a good idea to check valve clearances anyway.
Short open pipes without mufflers will screw up the jetting and exhaust resonance, so fitting a decent exhaust is always a good place to start.
And that brings us to other things that need to be checked. Start with giving it a complete service. Fresh oil, new filter, fresh spark plugs. Check the HT leads and caps because they break down with age. Testing resistance is easy. Ask if you are not sure how to and get a Factory Service Manual. They are available for download.
If the bike has sat since 2014 there is about a 99% probability that the fuel system will need to be cleaned. If the gas is old, dump it and flush the tank out. Clean out the fuel tap and filter. And then clean out the carbs. It is quite likely that some of the jets and passageways will be partially or completely clogged. That sounds like a daunting task but it's not as hard as it appears. Just read the FSM a few times and approach it methodically and ask questions as you go. If the prior owner emptied the tank and drained the carbs, you might get lucky with fresh gas and a can of Seafoam, but don't bet on it.
This link
http://pinkpossum.com/GT750/carb/BS40carbs.htm is for a GT750 Suzuki two stroke, but the carb design is the same as yours. The jets may be different but it gives you some idea as to what is what.