I have seen staff send in a letter like that.
Management should then communicate back, if they disagree with the proposed dates and/or final paid week’s leave suggestion.
ACAS is free.
Absolutely no need to spend money on a solicitor on this.
If it comes to it, a money claim online (via HM Courts, NOT using solicitors craftily copying their web interface) will cost peanuts.
ACAS give great free advice over the phone even if you have no union connections, just be patient with the phone queue. Their team are highly skilled on all aspects of employment law, and will tell you exactly what to do.