A private customer being a non-VAT registered party can expect the invoice presented to be the full and final amount. VAT, where charged is deemed included since, for the private customer it is not recoverable. If the supplier is not charging sales VAT through being under the threshold then the customer is expected to pay only the amount invoiced. The supplier in this case is expecting his customers to pay for his good fortune in getting more work but bad fortune in going over the threshold. That is the supplier's poblem. The customer got a price and paid it. End of story.
Interestingly , as Davyjones mentioned, if this were allowable, the original invoice would have to be adjusted anyway as it would have included VAT on the materials that the supplier was passing on from his purchase and once registered he could recover that and only charge the customer the low rate ( assuming the 2/3rds rule was met) that would result in a revised invoice which would not be as high as just adding VAT onto it all. However that is moot since the OP has no obligation whatsoever to pay anything more. ( and that would be the general law in either ROI or the UK as regards VAT )