I'm not sure if this is a restrictive or a non-restrictive clause (using the word 'whilst')

I’ve been tasked with editing a new hire’s bio, and I’ve been stumped by comma. For context, the intern is writing about how they found the company’s job ad. They are currently still studying at university. Here’s the original:

I was looking to gain some work experience , whilst I’m studying,
and I happened upon a [company name] post advertising an internship
position.

Here’s my potential edit (without the comma):

I was looking to gain some work experience whilst I’m studying, and I
happened upon a [company name] post advertising an internship position.

Am I overthinking this? Is there no need to remove the comma?

Edit: I forgot to specify that this is UK English.