No. A commercial invoice or statement of charges from a freight forwarder cannot be used to support an input VAT reclaim. HMRC requires either an official C79 certificate (for tax paid at the border) or a Monthly Postponed Import VAT Statement (for postponed accounting).
If PVA is not elected on the customs entry, the import VAT must be paid upfront before goods are released (or through a deferment account). You must then wait for HMRC to issue a C79 certificate to reclaim the tax on your return.
HMRC makes Monthly Postponed Import VAT Statements available online for only six months from the date of issue. Businesses must download and store electronic copies in their permanent tax archive to support future audits.
A non-U.K. business can only use U.K. postponed VAT accounting if it holds a valid U.K. VAT registration and U.K. EORI number, and is declared as the importer of record on the customs declaration.