Well you've lost a week already.
As I said on Page 3, there is very little to be gained by getting a Gas Safe inspection. They can put a rectification notice on your hapless installer but he is clueless. Having seen the job they will no doubt start looking at his work more regularly - but judging by the pushfit and compression fittings, he's a bathroom and kharzi fitter. He might only do one gas job a year, and after GSR have laid into him, that number might reduce.
But in the meantime, you have a potentially dodgy boiler and a baby imminent so I would learn a good lesson, and pay the extra £500. If you can find someone to touch it.
If we were asked we wouldn't want to get involved, you would have to take the boiler off the wall again to be sure and there are too many risks (eg: possible broken boiler pressure relief spigot, missing screws) that would take our time spent beyond the £500.00 value.
Most installers don't want to take on a cowboy half finished can of worms, because it isn't possible to be absolutely certain how far the remedial work will go and therefore be sure of making a margin. For those of you who don't work on the tools, 'margin' is 100% of your salary.