Funnily enough I have a client like that at the moment... *sigh*
Some times I wonder how they managed to make it this far in the business world. My client likes to dictate horrible designs to me, then after a few stages he will ask me what I honestly think *he must know it looks terrible....) and I have to tell him... then it's back to the drawing board. Well that's what I did at first, now I just tell him it looks fine, because otherwise I'm not going to get paid. To agree with and build on Levi's point, before I started having clients I used to think about how great it was going to be to create perfect logos, do amazing work and get paid for it! Now I have come to realise that actually, it's a very difficult game to play some times and if you don't handle situations correctly in an efficient manner then you will quickly become frustrated (case in point) and be out of pocket.
In your situation I would personally follow Greg's advice. Find out why he isn't particularly fussed about his website and then tailor your response based on his answer. If, for example, he simply doesn't think his website needs to look good, I would use a comparative analogy and explain to him that websites are more than just a sales platform, it's an extension of their company and more importantly their virtual shop front. If you walked into a physical shop and they had post it notes for price tags, the shop was a mess, all the shelves and aisles were horribly mis aligned, floor tiles were missing etc etc you simply wouldn't want to shop there, it's the same for websites.
As for showcasing your work, just make your own backgrounds to put the logo on, I like to make a background collage including smaller parts of the logo in the background and then some larger sections in the foreground with the logo in the centre. It looks professional and you don't have to worry about people worrying about the quality. You can see that image on my (still unfinished...) website:
www.ladesigns.co.uk it's the huge box on the left.