You could try suggesting the contract be amended to change the tenancy start date to 4th. Of course this may not work for the tenant if they need to move on 1st. Alternatively you could suggest an addendum to the agreement creating an initial period of 1st to 3rd and then regular calendar monthly periods thereafter from 4th to 3rd of each month. The tenant would then only have to pay the first 3 days of rent before moving in, (the “initial rent”), which may be doable. He would then pay a full month’s rent on 4th of each month. If you do this, make sure you follow the process outlined in section 8 of the Renters Rights Act.
Ultimately it’s not clear whether you can withhold possession until the tenant pays once the contract has been signed. Opinion I’ve read is divided on this point, with some very prominent lawyers seemingly unable to give a definitive answer.