There's nothing to say an employer has to allow you to use holiday to reduce your notice period.
For an admin role, four months is very long. For a potential new employer, say their existing admin person gives a month's notice. They start the recruitment process for a replacement, and are, with luck, in a position to make an offer within a month. If the potential new employee is on a month's notice they'll have to wait another month, ie a month without admin support, which especially for a small business, who might only have one admin person, not a team, can be very difficult.
If they are waivering between offering the job to someone on a month's notice or less, or someone for whom they'll have to wait another four months, ie a total of five months with no admin support, it will be a factor in their decision, and it doesn't show them to be disorganised or unstable.
I wouldn't accept four months OP. If you want the payrise and are minded to accept, I would suggest offering three months instead.