Yes the company can use its funds to buy a van. AFAIK VAT on commercial vehicles can still be offset against VAT receipts (check this with an accountant). Private HP agreement would not be advisable as this would create problems in applying the payment for the van against company funds, amongst others. Why not let the company enter the HP agreement? You can supply a private guarantee if required.
Loan would not be an issue, provided bank was happy to facilitate same. No issue on using company funds to cover loan repayments.
BIK would apply if the van was habitually used for personal use. With commercial vehicles it tends to be a grey area & unless you have clearly no other vehicle available for own use, it doesn't tend to be an issue.