This option would require some reformulating in your translation, but it is the term commonly used in the UK for the communication of academic research beyond universities.
'Impact' is another commonly used term which could be of use elsewhere in the text, i.e. public engagement is a means of achieving economic and societal impact.

...to capture knowledge transfer in a publicly transparent way...

The German, in a typical roundabout way, sounds as if knowledge transfer were already happening in the public domain and now one only has to go out and come up with effective ways to 'capture' it (presumably make it more accessible)

....to capture publicly shaped knowledge transfer

My translation of the whole phrase would read:
"... innovative methodological approaches to conducting surveys of knowledge transfer which has been made publicly available."
or, more succinctly:
"... innovative methodological approaches to conducting surveys of publicly available knowledge transfer."
I would not translate "erfassen" as "capture" in this context. Here it means "to survey", "to analyse", "to assess", or "to measure".

Basically they are wondering how this knowledge transfer into the public realm is happening in specific in order to optimize and utilize the process in the future. Thus one could talk about "making something public" or "disseminating" maybe also "publishing" information in general and omit the gestaltet.

"Öffentlich gestaltet" here means distribution via easily accessible channels, including viral distribution as opposed to established insider pathways reaching only a limited audience and effect.
The expression is a little misleading (or suboptimal) in the sense that öffentlich gestaltet expresses that the public or said "Akteure aus Wirtschaft, Politik und Zivilgesellschaft " play an active role in the dissemination of the information and may even choose to focus only on certain aspects of that information. I believe, that this is a given in every process of information dissemination though. It is always a chain reaction of some kind.

Therefore I suggest to focus on finding a translation for the whole sentence rather than öffentlich gestaltet in particular.

e.g.:
...und erfordern innovative methodische Ansätze, um öffentlich gestalteten Wissenstransfer zu erfassen."

..and require an innovative, methodological approach to record/ understand/ investigate/learn how knowledge transfer to the public domain takes shape./is happening in detail.