Wait, so do you have a swamp cooler? As in, a bucket for submerging your fermentor that you can fill with water? Because those things are pretty good at keeping the temp down, especially when it's only the 2-5 degrees C you're talking about. Remember, the first three days are the most important, after that it's okay to let it rise. I know it's a lot of work, and it sounds like you're pressed for time, but that's the way it goes.
The thing about English yeasts is they already put out a bunch of crazy flavors. If you ferment higher, I can really see a situation in which you make something undrinkable that you hate - estery to the point of gagging, a lot of diacetyl, spiciness, gourd/melon-flavored fruitiness... it scares me.
I use 1968 in most of my ales because I love its malty, estery profile, but I've even started cutting back and pitching around 62, letting free rise to 68, and finishing out at 72. The bulk of fermentation occurs around 65. My beers are still quite estery.
What about trying this: do you have the ability to chill the wort down to a pitching temp of about 16 or 17C? If so, you might get a little better fermentation early on before it gets too hot.
And I'm sorry, I just can't resist hammering the importance of fermentation temps. Forget all the science, all the reasons why - I experienced an ENORMOUS increase in the quality of my beers when I started controlling temps. It's the biggest single improvement I've ever made to my brewing... It's worth not brewing a couple batches, saving the money, and buying something to make your beer better. Good luck.