First off, I agree with Phil that (acquired) distinctiveness is the term of art and there is no justifiable reason for changing it. Also, from what I’ve seen, poll is much less likely to be used in this context.

Let’s examine the German first. We already have a translation for the last bit, so the question is what to use for Rechtsdemoskopische Gutachten.

Many sources will say demoskopisch, not rechtsdemoskopisch. A Gutachten is a widely used method for proving something in a (civil or criminal) trial but is little more than a survey, or Umfrage, in this case. See the last page at http://www.markenverband.de/veranstaltungen/Archiv Veranstal...

Cf https://www.booklooker.de/Bücher/Rainer-Utz Die-demoskopisch...

The authors of a Gutachten may explain the survey methodology and results, but this hardly rises to the level of a legal review. One of the organizations offering demoskopische Gutachten is IfD Allensbach and their EN page says "surveys for legal evidence": https://www.ifd-allensbach.de/legal-opinion.html

The above phrase can be found in ENS books, though I think survey as evidence, or survey evidence, is more likely. For example, see "The Trademark Clarification Act of 1983," printed for the use of the Committee on the Judiciary, or the following link:
"Consumer surveys as evidence of trade mark infringement

…In the UK, survey evidence is often used to demonstrate a likelihood of confusion in a trademark action or that there is a misrepresentation or a likelihood of deception in a passing off action. For example, survey evidence was successfully used to support Associated Newspapers’ claim that the publication of a paper called Evening Mail would cause confusion with their titles…"
https://www.twobirds.com/en/news/articles/2010/master-of-all...

Even in German (and in the text you quoted), "demoskopische Gutachten" are used "als Beweismittel," as stated in "Wettbewerbsrecht": https://beck-online.beck.de/?vpath=bibdata/komm/GloyLoErdHdb...

As said (and agreed) in the discussion box, the best option you have in English (aside from using some lengthy and complex solution) is to rephrase based on the sentence in which the words appear. This is because "auf Basis von Umfragen" is redundant, or, more precisely, an in-text explanation of "demoskopisch" (which you don't translate), and "Beweismittel" shows up in the first sentence, too.

To make a long story short, you could simply break up survey evidence and use either "survey as evidence" or some variation thereof.

Furthermore, "...survey to establish..." is common enough, in my opinion, whatever the context. An entirely different example from another ENS document: https://howlong-p.schools.nsw.gov.au/content/dam/doe/sws/sch...

I believe the most important thing is to use a descriptor that captures the essence of what is being said, without being too wordy. People will know what you mean anyway. This is why you could also call this a trademark survey, as the following does:
"Surveys to prove or disprove trademark infringement or likelihood of confusion have been used by attorneys for many years…There is a commonly held belief that trademark surveys are terribly expensive. This is not necessarily so.”
https://www.ipwatchdog.com/2019/03/02/ten-things-avoid-trade...

As a book title: https://store.lexisnexis.com/products/trademark-surveys-a-li...

Mine is pretty close to the description in the U.S. Patent Office’s Trademark Manual of Examining Procedure (maybe bookmark the manual for future jobs):
"Establishing Distinctiveness by Actual Evidence
…Survey evidence, market research, and consumer reaction studies are relevant in establishing acquired distinctiveness and secondary meaning."

Additionally: "The kind and amount of evidence necessary to establish that a mark has acquired distinctiveness in relation to goods or services depends on the nature of the mark and the circumstances surrounding the use of the mark in each case."
https://tmep.uspto.gov/RDMS/TMEP/Oct2012#/Oct2012/TMEP-1200d...

As an aside, the above doesn't mean that survey evidence is unheard of in the UK: https://www.gov.uk/government/publications/mergers-consumer-...