As far as I can see it is perfectly legal and there’s no reason you shouldn’t do it if you want to. The trust can’t make you work 5 consecutive nights, though.
The Working Time Regulations are a good starting point for determining what is safe working. They do not limit the number of nights you can work in a row.
The 2016 contract is a document negotiated between the BMA and NHS Employers. The “4 shifts in a row” limit is in there as an employment condition - just like your salary, annual leave entitlement, etc. It does not imply anything about the safety of working more than 4 night shifts in a row.
You and your employer can vary the T&Cs if you both agree.