The first step must be to determine who the software was actually being supplied to. Was it bought by the director as an individual consumer, or acting on behalf of the company? If it's the former, the director received the supply and made an onward supply to the company. As the director won't be VAT registered, there's no scope for VAT recovery by the company.
Had the original supplier been UK based, it might not have been clear what capacity the director was acting in. Especially for lower value invoices, which don't need to show a customer name, HMRC are likely to accept that the purchase is on behalf of the company unless there is anything which suggests otherwise. But when buying digital services from a supplier that is not based in the UK, the UK VAT charge makes it clear that they considered that they were selling to a consumer rather than to a business. HMRC might accept VAT recovery by the company if they're convinced that the director was buying on behalf of the company and failed to provide the VAT registration number in error - but it's risky.
If the director really was buying on behalf of the company, I believe that the correct (risk-free) approach would be to go back to the supplier. Explain that the purchase was made on behalf of the company and provide the company's VAT registration number. They may be prepared to issue a credit note to the director, an invoice without VAT in the company's name, and refund to the director the VAT previously charged. The company would only need to pay the net amount to the director, and could do the reverse charge entries on their VAT return using the new invoice from the supplier. But this may not now be practical, and is reliant on the supplier agreeing.