No, I don’t think that you could do this with you as sole trader for both businesses. HMRC would likely regard it as a single business in reality. Also if a business is registered for VAT, it must charge VAT on its sales – this can’t be waived by it failing to reclaim VAT on its costs.
You could VAT register voluntarily (you don’t have to wait until your turnover reaches the £85k) – but the cost of charging VAT on your sales would outweigh the benefit of reclaiming it on your costs.
Do you know why the auction site are insisting on this? Is it just a way to ensure that they are dealing with businesses – if so is there any other proof that they would accept? There’s bound to be others in the same position as you, and I can’t think why they would want to restrict their potential customer base in this way.
Someone else with a VAT registered business could always buy from the site and sell to you “at cost”. That business would reclaim VAT on the purchase and declare VAT on the sale to you – so no overall impact on them.