Recognise your role in the organisation
I think the first thing to cover is the separation of roles here: you are a web developer working with a web designer, to a certain extent your role is to implement the design as requested by the designer and approved by the manager. As such, approaching management and going around the designer will not look good for you, as you are attempting to work outside of your domain.
It may also be that, while you have disagreements, the designers work may actually have a rationale behind it. What would help here is increased communication between you both. If the designer is recommending a giant button, find out why. Is it to improve user click-throughs. However, if they are unable or unwilling to explain it, don't press the issue. As stated, your job is to implement their design.
Ask, don't tell
If they do give reasons behind it, and you think there is a better approach, ask don't tell. Ask if they have considered other approaches, ask if you can help with the design process to expand your own skills, ask if you can see the process they went through to come up with the final design, ask if they can critique some of your own ideas and help improve them.
In the best case, you will get an understanding of why they have taken the approaches they have, in the worst case you might gently give them a reason to provide rationale to themself for their ideas.
Recognise your own potential limitations
Sometimes, when working across domains, we can over estimate our own skills, this is called the Dunning-Kruger Effect. User Interface and Experience is such a broad and difficult field, and is much more nuanced than people recognise. Thiscan be seen by the fact that there is a User Experience Stack Exchange site, that is very separate from StackOverflow itself despite often coving similar technologies. While you may have a lot of experience building websites with these technologies, it is worth examining the possibilty that there are gaps in your knowledge that the designer is exposing that is causing some friction between you both.