Is Exercise Before Tefillah Permissible?

I run regularly, and have been logging somewhere between 12-15 miles per week for years now. (I'm almost reached "purple" on NikeRunning - about 30 miles to go!) I run to try and keep a handle on my weight, but even more to keep myself sane. Running helps me stay balanced and in control, and is something I make sure to keep doing on a regular basis.
The problem, especially in the summer, is that it's hot. Really, really hot. And in Yad Binyamin, which is essentially the coastal plane of Israel, its especially hot, reaching the mid-90's (F) regularly during the day. You can either run late at night, or even better, early in the morning - very early, which I've been doing for much of the summer. But you have to run really early, before the sun gets too high in the sky and the air grows too hot. By 8am, it's just too late to run at all during the summer months.
So I've been getting up early - a little after 6:00am to run. The air is just cool enough to breathe, and I can cool down and shower and, if I'm lucky, make minyan by 8:00am (and if not, there's always 8:30).
Then, a couple of weeks ago, a member of the community approached me with a question. He too needs to work out regularly - although he prefers swimming. (It seems that when he lived in Gush Katif he would snorkel regularly in the early morning in the ocean and would allow himself to do so with the rationale that the ocean is essentially a mikveh, and his immersion is halachically acceptable before tefillah. But now, his swim would be in a pool and wouldn't count.) He noticed me running in the morning, so he wondered about the halachic basis for my practice. Why would I be allowed to run before davening?
His question becomes even stronger in light of a statement of the Gemara from this week's Daf Yomi where the Gemara (Brachot 14a) states:

אמר רב אידי בר אבין אמר רב יצחק בר אשיאן אסור לו לאדם לעשות חפציו קודם שיתפלל שנאמר (תהילים פה) צדק לפניו יהלך וישם לדרך פעמיו: 

R. Idi b. Abin said in the name of R. Isaac b. Ashian: It is forbidden to a man to do his own business before he says his prayers, as it says, Righteousness shall go before him and then he shall set his steps on his own way.

It's pretty clear from the Gemara and the halachic rulings that follow, that a person must place prayer as the first priority in his daily schedule. First pray to God and then do everything else. The Shulchan Aruch rules:

אסור לו להתעסק בצרכיו או לילך לדרך עד שיתפלל תפלת שמונה עשרה (ויש מקילין לאחר שאמרו מקצת ברכות קודם שאמרו ברוך שאמר וטוב להחמיר בזה) (תרומת הדשן סימן י"ח) ולא לאכול ולא לשתות אבל מים מותר לשתות קודם תפלה בין בחול ובין בשבת ויום טוב וכן אוכלים ומשקין לרפואה מותר:

A person is forbidden in engaging in his needs or traveling on the road until he prays Shemonah Esreh (some are lenient after he recites some of the brachot that are said before Baruch She'amar, and it is proper to be stringent in this matter - Terumat Hadeshen 18) and not eat or drink. Yet, [drinking] water is permitted before Tefillah both during the week and on Shabbat and Yom Tov, and eating and drinking for medicinal purposes is permitted.

The ruling seems rather strict: no eating, drinking - not even the study of Torah is allowed - before one prays to God.
Indeed, to me the ruling makes intuitive sense. A religious individual starts his or her day by first acknowledging God's presence and influence in her life. That faith and devotion sets the critical tone for the rest of the day, and is a cornerstone of a spiritual life. What we do first, at the beginning of the day, says everything about us: our priorities and attitudes about the way we live our lives.
With this background, is there any wiggle room to permit running, or any type of exercise, before engaging in daily prayer?
Actually, there is. Piskei Teshuvot, a compendium of responsa written pretty much over the last century, has a nice summary of the various reasons that a person would want to engage in all sorts of activities before prayer:

• Can you engage in tzedakah before davening or acts related to Chesed? Yes?
• What about helping get your kids out to school? Rav Shlomo Zalman Aurbach allowed it.
• How about buying minor food items (i.e. milk and bread) to get said kids out to school? Again yes.

So, there is some amount of wiggle room to allow for a certain level of activity before prayer. What about running and exercise? Piskei Teshuvot (O.C. 89:15) quotes Rivevot Ephraim who likens exercise to eating for medicinal purposes, which the Shulchan Aruch explicitly allows. He allows "running for medicinal purposes or exercising in the water, for someone who requires this activity for medicinal needs, and anything similar to this..." (see footnote 190) He also quotes Sefer Tefillah Kehilchatah in the name of Rav Shainberg who allowed exercise for health, explaining that this is no worse than eating or drinking which Shulchan Aruch allows, but only "if there's a justifiable reason" for exercising before prayer.
Moreover, one can and should combine this with the leniency of Rema, who allows one to engage in forms of activity after reciting birchot hashachar - morning blessings. Then, you're combining the fact that you have indeed offered words of prayer with the leniency of health concerns.
This leaves us with two questions: is it absolutely necessary to run before davening? Theoretically, I could get up and daven vatikin, at sunrise, and run afterwards. That's hasn't really happened thus far. Is my inability/unwillingness to get up at 5am to run a mitigating factor which will allow me to run before I daven? Secondly, all of the sources discuss exercise for "refuah", which generally refers to someone in rehabilitation, or someone who requires exercise on the advice and counsel of a doctor for a specific malady. While I don't know any doctor who would not advice regular exercise, does my need to run regularly rise to the level of "refuah"?
I think so, but I could see someone disagreeing.
In any case, hopefully the staggering heat will dissipate soon and I can get back to running at a normal hour.