I went for another agency role a couple of months ago which was through a recruiter, so I discussed the part-time thing with the recruiter first and they discussed it with the employer before putting me forward. I still got an interview but didn't get the job.
I think if I interview at another place for something that's advertised as full time, I wouldn't bring up part time unless I really had to in the first interview. I'd wait until they offered a second interview and then say something like "I'm really excited to explore this role further as it sounds great. I want to let you know I am available to work 4 days per week and believe I can be very effective in this role at the reduced hours"... Or something along those lines depending on the situation. I think most employers have made a decision on who they want after first interview and second interview is more of a formality, so if they are keen they may be willing to consider part time. If they say no, it's not the right role.
I work in digital marketing and am client-facing. The challenge I have had is that lots of agencies have the mentality that client-facing roles have to be full time, which is rubbish really. It's all about managing expectations and having the right processes in place. I have always been happy to take calls and check emails on my days off for anything really urgent, but generally it's completely workable. Unfortunately my industry has a bad track record of working people into the ground and expecting way too much commitment. Most people burn out, change career paths, or go client-side at some point in order to get their lives back!