What can I do to get rid of smells from bathroom plumbing?

something must be there. I would contact people that service your water filters and not try any homemade remedies. That may cost you a lot of money if whatever you do will damage your system

You might need an "S" trap that can hold water, so odors from the septic do not come up the drains, same with tubs and showers

Does the water smell in other locations or just the bathroom sink? If only this one location you need to clean the drain out. You can approach the job from either direction if you do it manually. From the top use a stiff wire with a slight hook on the end. Run this down the drain, then twist it, then withdraw it, pulling out hair and gunk. If you have the white plastic pipes you can unscrew the joint and drain the water into a container, then remove all the gunk...and said gunk will be nasty and smelly. Or use a drain cleaner. Pour this into the drain at bedtime and rinse the drain out the next morning. You might need to do this two nights. Then use paper towels and a household cleaner in the drain, then sticking paper towels into the drain and turning them. You might also remove the aerator and clean it with a toothbrush. Good luck.

I agree with Suellen, it maybe just a matter of a dirty drain with gunk in it. To start with, if you have not already done this, remove the piece that sits over the top of the drain that stoppers the sink, then add 1/2 cup baking soda and add 1 cup white vinegar, when the fizzing stops rinse well with lots of hot water. If the drain cover is dirty-the one you removed and extends in to the drain-clean it with soapy water and an old toothbrush.

If still smelly, put a plastic container under the trap and remove the trap under the sink, if any contents are in it then clean them out manually and finish cleaning this trap in another sink.

You do not need to shut the water off for this procedure.

There are drain cleaners-available at diy stores that want you to let them soak undisturbed for at least 8 hours-or overnight, just follow their instruction on the container.