It's not an area of VAT that I have much involvement in, but here are some thoughts in case they help. I am assuming that the company you're dealing with is a principal in the supplies - i.e. they're not an agent in a supply by you to the person to whom the goods are being exported.
I think that it depends on whether your customer has a business establishment within the UK - see notice 703 final paragraph of 2.9 and 2.10.
If they have a business establishment in the UK, it's not an indirect export. You are making a standard rated supply to them, and they (subject to meeting the normal conditions) are making a zero rated export to their customer. This shouldn't be a problem for them, as they would be entitled to recover the VAT you charge via their UK VAT return, due to them then making an onward zero rated export.
However, having a UK VAT registration doesn't necessarily mean that they have a UK establishment - non-established taxable persons (NETPs) are required to register too. See https://www.gov.uk/hmrc-internal-manuals/vat-registration-manual/vatreg37050. Might be worth looking up their VAT registration number on the HMRC checker https://www.gov.uk/check-uk-vat-number - this may help in determining whether there is a UK establishment?
So considering your three questions, my opinions would be:
1. Yes, if your customer has a UK business establishment and you are not the exporter of the goods, you should charge VAT.
2. No, I don't think that having proof that your customer has then exported them impacts on the treatment of your transaction. That's because if it were to impact, how many steps back through a supply chain might theoretically be affected!
3. As you're not the direct exporter, zero rating would only apply if the criteria for an indirect export were met. If the customer has a UK business establishment, they're not met, regardless of whether you obtain a foreign VAT number from them.
I'm still struggling to see the benefit from the customer's perspective, apart perhaps from cashflow if they need to pay you the VAT significantly earlier they can recover it on their VAT return. If they're still convinced that VAT isn't due, perhaps you could refer them to the extract from notice 703, and ask them to provide extracts of guidance that supports their view?