But, Jake, any “constitution” will only be manipulated by the sitting government and will effectively make little or no difference to the policies they pursue.
All the matters you mention can be altered by the two Houses of Parliament. They will continue to be so with or without a constitution. The specific matter of what happens in the event of a hung Parliament is quite clear. The Prime Minister, who is not elected by the populace, remains the PM until it is clear he cannot command a majority in the Commons. In the event of a single party gaining an overall majority this point is reached quite quickly and obviously. In the event of a hung Parliament, it takes more time and that is precisely what occurred.
The “constitutional” issue that is most prevalent in the UK is that the needs and requirements of the UK electorate which should be determined by their MPs are being unnecessarily modified or even abandoned to accommodate the legislation and rulings that stem from Europe (either the EU itself or the ECHR, which I quite accept are separate). This is being done, ostensibly, to homogenise and harmonise procedures throughout the EU.
Which is all very well, but very often these measures are contrary to the wishes of the UK electorate and their MPs. Such phrases as “...we have no choice but to comply...” and “...EU regulations do not permit...” are simply not the things that people want to hear from their elected leaders. We are not, at present at least, a single European nation and the people of this country expect their leaders to act in the best interests of the UK. Whether or not that complies with the wishes of our “partners” or whether it upsets them should be of no relevance or concern.