Apply for child's first passport

[mtalhakhan15]

Hi respected gurus, mods and members,

I got my ILR in May '21 and soon after applied to register my son as a British citizen who was born in the UK in 2020. My son's certificate of registration arrived this week.

As we weren't sure how long it was going to take to get his registration certificate, we booked travel to Pakistan more than 3 weeks ago. Now the dilemma is applying for his passport as we would also need to submit his current foreign (Pakistani) passport with the application. Due to the tight timescales and work/personal commitments, I can't even use the 1 week fast track service and there is no point applying before the trip as we won't be able to travel without his PK passport. Based on my research, I believe I have the following 2 options and it would be good if someone could confirm if either (or both) are fine:

1) I apply for a British passport overseas from Pakistan.
The only confusion I have with this method is that in the supporting documents, they mention "One of the documents provided should show a link to the parent applying and show that the child and parent are resident where they are applying from". Neither my wife nor myself are residents in Pakistan as both of us live in the UK and are only going to Pakistan to visit family. We can provide the birth registration certificate from the UK showing our details but not sure how to address the 'child and parent are resident where they are applying from' part.

Should I follow guidance for 'applications from overseas' as much as possible and attach a cover letter explaining why I can (and can't) provide certain documentation (E.g. Residence in Pakistan) and apply from Pakistan once I am there?
The link to the supporting documents can be found here.

As a comparison, I can easily provide all the documents needed if we were to apply from the UK. This overseas application seems to be quite complicated!

2) I am planning to come back a couple of weeks before my wife and my son. In this scenario, I come back and bring my son's passport with me and apply from the UK using the Fast Track service. Once I get the passport I can courier via priority service to Pakistan and my wife can receive it there to use for travel to the UK. The only question in this case is, would it be an issue if my son is not in the UK when I apply as his passport stamp will also show entry in Pakistan but not exit?

Any help, advise and guidance to my query is much appreciated! Thank you.