I get my prescription from my primary care provider during my annual physical. I don't understand the question about what to say if you don't really need it, I just tell them that at times it doesn't t respond as well as I'd like, I sometimes lose my erection and I'd like some help with that. Now, I'm in my 60s and the statement is true so it's not a tough sell, but it's not like they bring a nurse in to give you a blow job to prove you can't get it up. Even if you're younger that doesn't mean it couldn't be true, tell them you think it's job stress, you're kind of embarrassed to admit it at your age but you'd just like a little assistance once in a while. I think you're going to find it's not a big deal, it's a doctor, they hear these things.
The other thing I do is tell them I'd like a separate paper script rather than having it transmitted electronically with my other prescriptions (blood pressure, etc.) as I looked into it and I can get it cheaper at Costco than through my insurance. That keeps it separated and I pay cash at Costco and it doesn't show up in my insurance statements or credit card. I also use GoodRx at Costco and it is pretty cheap, especially compared to what it was before generics were allowed. Funny thing happened last time I got it renewed, my doctor has always written it for six 100mg pills with six refills, kind of a pain but it's not like I'm refilling every other week. So this last time the pharmacist told me that it would be cheaper per pill if they just sold me a prepackaged 30 pill bottle, and asked me if that was OK. So 5 pills with 6 refills will be 35 pills instead of 30, but it would be easier than going back for refills so I said sure. And when I got it I found that they'd marked it as having 6 refills, I don't know if that was a mistake or they just aren't that worried about it as long as you've got a prescription, but holy crap that would be a lot of viagra (actually generic sildenafil).
As far as the piss test, I'm not a doctor or lawyer and I don't play one on TV, but a little reading on the interwebs would seem to say it's probably not a problem. First question is whether the concern is work/security clearance or for sports. It seems like it typically leaves the body within about 6 hours, and for work there shouldn't be any reason they're screening for it, it's not an illegal subtance and you're getting a prescription so even if it shows up who cares. Not their business. For sports there seems to be some thoughts that increased blood flow might have some slight performance enhancing affects, but that doesn't appear to be a serious enough concern that it's been labeled a banned substance, so I don't think it's even being screened for. Recommend you do your own reading and make up your own mind.
Hope this helps.