He's our biological child and he has his own passport. We'd like for all three of us to travel together but it may be me & my son traveling separately from my husband. Our goal is to travel together, though.

We called the Qatari Embassy and spoke with a rep in visa section and he said that our son will need to be added to the letter of employment. He wasn't very helpful beyond that, unfortunately. And we didn't want to raise the flag by asking him how we can circumvent the proper channel of bringing our son.

If we were to do what you suggest, wouldn't the consular official be suspicious of the fact that my husband and I will be travel on business visas but our son on a visitor's visa???
So is it easy to obtain residence permit? What are the requirements?