There is probably no solution if the management does not want to move from status quo.
If they demand to create wrong invoices then they must accept cancellations or allow to wait until weight is known to create the invoice.
Alternative you can of course create all kind of virtual documents and movements, e.g. a second delivery and second invoice in case the physical weight was higher than the billed.
Or a virtual return with a credit note if the weight was less than the original billed weight.