Try to get an email from your HR stating they cannot give reference letters for your application. You should add this email in your letter of explanation stating company refused to give the reference letter in the format you want.
Get whatever letter your HR is willing to give - with as many details as possible - and add that in the pdf for this particular job.
Add all official documents (offer letter, paystubs if you have them, maybe a tax document which shows you’re working for this company etc.)
Get letters from colleagues/supervisors with the job duties and responsibilities written that conform with your NOC code.
I’m not sure about this affidavit thing; I did it on plain paper and it worked for me. I don’t know which consultancy you’re quoting, because not a lot of ppl I know who apply for PR go through any immigration consultancy; they’re obviously trying to play it safe which is a good thing, but if you’re in crunch time you need to also use the fastest way. If you don’t get the said affidavit in time and fail to submit it it’s 100% reject/losing points for that job. With the approach I mentioned it might just work.
I don’t know if CIC has changed any rules about submitting proof for companies not willing to give letters, but the affidavit from judge etc whatever you mentioned sounds too complicated for EE application.
If you consultancy says so, try to get this affidavit (maybe from a notary?). In the meantime try to collect documents related to this job and all other documents for your application in time.
CIC is usually considerate of such cases.
Shooting in the dark here; but if you work for, say Infosys, they have a portal where you can submit requests for reference letter. They send you a draft and they’ll revise it and once it’s finalized they’ll get it signed from the manager; that’s how I got mine (very late, after I had received my PR ).