No, they cannot test it through urine. Your urinary system and your reproductive system (vagina) are two totally different things. Bacteria in urine would mean a urinary tract infection. So, the doctor woul have to do a pelvic exam, which would in fact require your “privacy” to be invaded. They will take a sample from your vagina and look at it under a microscope. If it is BV, you will either be given a prescription medication taken orally or one inserted vaginally. I know you said you’re not sexually active, but just as a precaution, if you use the vaginal medication, you will want to remain that way. The oral medication can cause a metallic taste in your mouth and you absolutely will not want to drink any type of alcohol while taking this medication. It causes severe side effects. (I know you said you’re in hs, but you never know now days).
Also, just an FYI, BV is not a sexually transmitted disease. It is formed because your vaginal pH gets off balance and the good bacteria in your vaginal area get overrun with bad ones (to put in easy terms). Eating yogurt regularly (no sugar added), avoiding sugars, avoiding perfumed soaps and lotions in that area, and wearing plain white cotton panties (or none at all) is helpful. Sometimes thong panties, or panties of the type, are not good for your “private” area because the string part that is around your anal area can come in contact with your vaginal area, which is a major cause of BV.
You will have to see your doctor, if you feel this is BV. I would suggest you see him/her regardless because any foul odor in the vaginal area is a sign of something being wrong. Only a prescription can fix BV. If it’s a yeast infection, you can take OTC meds, but you would be experiencing other symptoms as well.
Don’t worry about your privacy being invaded. I know it’s hard at first, but trust me, they see tons of them everyday. It’s just like looking at an arm or leg for them. It’s just a body part that needs to be medically treated. Just be as honest with them as you can be, give as many facts as you can, and find a good doctor you can trust. Best of luck to you and I hope you get some relief.

Having your “privacy invaded” is the only way to get diagnosed. They take a sample of your discharge and look at it under the microscope, and to my knowledge, that has nothing to do with urine.

If you don’t want to deal with that, you can use alternative methods, but the doctor might end up checking you anyway to confirm diagnosis.