Technical SEO considerations aside, first.
The answer depends (in part) on who their target audience is and where they are located. Are they targeting Parisians or Glaswegians, for example? As these are two totally different audiences with totally different expectations.
I'm guessing it's aimed at English speakers across the UK mainland, so something with mainstream appeal would be preferable to my mind. There are few peoples on earth more staunchly proud of their nationality than the Scots (and rightly so), but the current branding may be a bit of an irrelevance or source of confusion at best or a turn-off at worst - for some. Of course, the opposite applies if you are selling to Scots or selling authentic Scottish produce, but they aren't.
If they are using .scot or .scotland or some other variant (again, technical SEO considerations aside at this stage) there is some risk of confusion as people are still more familiar with the .com/.co.uk domains and you wouldn't want to lose potential clients to competitors with similar (and more memorable) domain names. (These .vanity type domain names are gaining traction though - albeit when the .co.uk variants are taken - you only need to Google any ad/media/seo agency to see a profusion of .buzz .agency .hipster - I haven't looked, but I'll wager that's out there, too!)
So, brand perception and confusion aside, we can look at technical SEO:
Is there any benefit to using a different domain suffix (user perception aside)?
- And, this may be the main thrust of your question.
No, as all TLDs are treated equally by Google. (See: http://searchengineland.com/google-explains-how-they-handle-the-new-top-level-domains-tlds-225671
So, there is no gain to be had from changing over - from Google's perspective.
However, If they are targeting French speakers exclusively then I see a case for ruling out on a .fr domain, especially if you have a local presence over in France. It goes without saying that you'd want the site presented in French first, too - but that's a given. a Google My Business listing to target local search etc would benefit, of course.
You could go as far as hosting the site in France to boost server response by a few milliseconds or so, but this is only applicable if you're actively seeking a predominately French customer base.
If you are thinking of switching domains or moving sites there are obvious logistical considerations (redirects, migration of content etc - depending on how you roll it out, so there is risk of losing some ground in the SERPS (even if you execute the migration flawlessly).
Of course, if my speculation is correct and you are targeting UK users in the main and you are gaining any kind of traction with the current site, you may be better to focus on delivering a better on-page experience - "Best places to visit in Dordogne" type articles, so you're catching prospective clients early into the buying process, broadening your site with engaging content, carrying out some content outreach exercises, promoting your brand on Social Media etc and stick with what you've got and there's no reason at all why you can't rank very well for any term you wish - you just need to but the work in to get the rewards. Just persevere, produce great content, keep your target audience in mind and you can beat the best - and you'll never be short of Scottish customers.
Bonne chance!