Ha ha, I've been round the 'exotic material' loop lately. We use lots of Invar for thermal stability and purchasing kicked up a stink because it's expensive, only certain machine shops will work with it to the tolerances we need and it has long lead time.
Fortunately, for most of the high ticket items they flagged we had tried not using invar and had done testing that showed the difference. Once this was established they were more willing to accept our arguments on other parts where we hadn't tried not using Invar but were able to do some ball park calcs etc.
Finally, for the couple of parts we thought were worth trying non Invar version of (legacy parts copied onto a newer system because they were 'common' rather than based directly on performance), the cost savings they showed were too small to justify the amount of testing required. This one sucked a bit as I'd spent a bunch of time creating drawings for lower cost version but c'est la vie.
On the tolerance issue, baffle them with math. Some time back I posted a spreadsheet for the hole position tolerancing per ASME Y14.5-M-1994 appendix B(2?). Some folks questioned why have a spreadsheet for such simple calcs, well this here is one reason. Being able to show those calculations to folks in a well presented format seems to hold more sway than some chicken scratch 'on the back of a fag packet' (as we used to say in the UK).
Aerospace and the like tend to be better at keeping copies of design calcs than some other sectors - heck in Aerospace they're often compiled into formal documents. This may be excessive for you but at least if you do calcs in some kind of log book then you can go back and look at them.
Hopefully if you're able to show them the math, or the test results a few times then they'll gradually learn that most of the time you aren't making this stuff up and they won't question it.
However, keeping design records such as tolerance calculations etc. for things like this - especially critical or non standard things etc. may be a very good idea.
Posting guidelines faq731-376 (probably not aimed specifically at you)
What is Engineering anyway: faq1088-1484