Can i register as a Sole Trader eventhough its a private limited company

My problem is, if i say its a private limited company, shopify is asking for companies house registration number.

But if i give them that and they start displaying it in my store or invoice, people can look up my personal information on companies house website.

Shopify doesnt ask this if i select sole trader.

Can i register as sole trader and still pay my sales tax through my LTD company?