TL;DR: I don't think that this would be OK. The best course of action would be to make your own designs (even if they're for example sites) and use those for your "samples". Your clients would get a great feel for how you design, which is what you want to showcase.
Let's examine what Fair Use entails:
Notwithstanding the provisions of sections 17 U.S.C. § 106 and 17 U.S.C. § 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include:
- the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
- the nature of the copyrighted work;
- the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
- the effect of the use upon the potential market for or value of the copyrighted work.
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.
Below is how I'd analyze these in your case. A common contract clause states that the design copyright is held by the designer and that the design is perpetually licensed to the company for their use. If this is the case, the copyright holder of the design is not the company, but the designer.
- Purpose and character of the use: Since you would be doing this to promote yourself, this would likely fall as "commercial nature" and fall against fair use.
- Nature of copyrighted work: The original work is creative in nature (as opposed to factual). This doesn't favor a fair use argument.
- Amount and substantiality of the portion used: Since you're only using a portion of one of the pages, this will probably weigh as fair use. However, I could see an argument that the design is the main selling point, and showing the design, even though it's only on one page, counts as replicating enough of the design.
- The effect on the market or value: While it would increase the market, it wouldn't do so substantially. Furthermore, it wouldn't devalue the copyright holder's work. This is probably in favor of fair use.
Overall, I think your proposed use case would not fall under fair use. In addition, frankly, if I were to come across a website that claimed my design as their own, I would strongly consider legal action.
Honestly, the best thing to do would be to create your own designs and showcase those. It'll take longer, but it'll be more worth it in the end. Your clients would see your work, what you're capable of.