How complex is submitting self-assessment tax for foreign income in the UK?

Is filing self-assessment in this circumstance enough of a pain that it's reasonable for someone who hates doing paperwork to just turn down the offer, or is it easier than I think it is?

It's not terribly difficult - you have to register for a Government Gateway ID in advance (this gets sent by physical post so make sure it's set up in plenty of time). Then you go to the Self Assessment pages on the Government site and fill in your details. The first part of the form (after your personal details) is a big checklist of which elements apply to you, and then it tailors the rest of the form from there to only include the questions that are applicable, so you don't have to wade through pages and pages of irrelevant stuff. Every question has reasonably clear guidance notes linked directly from it.

Once you've filled in the form it calculates all the tax owed and tells you the answer. You can then save your SA form and the calculation for your own future reference. If the tax owing is below a certain threshold (from memory about £3000?) and you fill in the form early enough, they collect the additional tax through PAYE by changing your tax code, or you can opt to pay it as a one-off payment direct to HMRC.

The current tax year runs 6th April 2020 -> 5th April 2021. If you fill it in online you have until 31st January 2022 to complete it, but if you want them to claim back tax through your tax code you have to fill it in by 30th December 2021.

One thing to be aware of is that once you've started doing Self Assessment for one year, they will ask you to do it again every year. Once you've had a few years with nothing that requires SA, they will fall back to simple PAYE again.

Would I have to pay all my taxes for the year via Self-Assessment, or could I pay my taxes from my regular UK job by PAYE as normal and only file Self-Assessment for the small amount of foreign income? Any suggestions appreciated.

You still pay your normal taxes for your regular job through PAYE. Your employer gives you a P60 at the end of the year which includes the details of your earnings and tax paid. You copy the details from that into the Self Assessment form and then it does all the calculations for you to tell you how much you end up owing (i.e. just the tax on the foreign income).