Interesting. I did not know that. Perhaps if this process continued and instead of having each borough/county have around the same number of people as the British system (eventually) evolved into, have the buroughs be mostly set in stone and have the force of tradition behind them so that it becomes very difficult to change them. Then, in order to make the system more representative, they'd cut the rural districts down to 1 MP and keep on increasing the urban districts as population keeps on increasing. Eventually you'd get districts with ginormous numbers of MPs. For a while this would strengthen party organization since, as you say, the parties would have to coordinate how many candidates to run and who should vote for each one but beyond a certain point it'd start getting really easy for independents to grab a seat of their own without party backing and it'd get harder to keep the local party machines together. For example if you're a wing of the local party in a big district that wants a specific person, why bother coordinating with the local party when you have enough votes to get your guy by yourself.
Similarly keeping around some rotten/pocket buroughs might delay partisanship since if they're under the control of some rich guy they're not really beholden to some ideological party.
Some other ideas:
1. Have one really really big party like the Mexican PRI, the Japanese LDP or the American D-R during their periods of dominance. Having one really huge party means that most important politics happen under its aegis so you don't get normal partisan politics.
2. Organize Constitutional Monarchy differently. One thing that makes parties form is when you have some controversial leader you get a party of guys who like that leader and guys who don't like that leader, even if the parties you get are pretty disorganized (like the American Whigs). If you can keep the leader neutral (like UK's queen) but with just barely enough power to keep any single other person being a powerful enough lightning rod, that might help with that. Perhaps if you made a Constitutional Monarchy in which the monarch's powers are more like a stripped-down version of the American president's powers rather than like the Westminster system. For example the monarch would appoint ministers, not anyone elected, but they'd have to be confirmed and the elected house would write laws but wouldn't be in charge of their implementation/enforcement. That's still fairly democratic but maybe getting that big of monarchical power and would delay parties at least a bit. But still, bit of a long shot...
3. This would require a POD way way way back but how about changing from a representative democracy to a proxy democracy. What I mean by a proxy democracy would be that originally everyone from a (small) social class would have the right to go to meetings of the national assembly and vote in person (sort of like in Rome). However, this would be logistically difficult so people could designate a proxy to go to the capital and vote for them with multiple people being able to designate the same person as their proxy. So for example if Lord Adam showed up at the assembly he'd be able to cast his own vote, but Lords Bob, Chris and Dan could make Lord Edward their proxy and then Lord Edward could cast four votes. Then later expand the franchise bit by bit until it finally gets universal. Elections would be then you going to the government office and registering who you want your proxy to be and that person would cast votes in your name (and in a bunch of other people's names) until the time came for you to re-register who your proxy is. That should retard party formation good and hard since the various politicians would all be competing against each other for people to register them as their proxies. It'd also be resistant to party formation for the same reason direct democracy tends to be since that kind of proxy system is closer to direct democracy than representative democracy is.