Hi Grundsey,
As Sully said, there aren't any compelling data to support resistance training recommendations here when it comes to your aortic insufficiency ("leaky valve") - especially data looking at real, long-term clinical outcomes, versus just measuring regurgitant fraction during exercise.
Most of the typical cardiologist recommendations you'll get are based on "physiologic hypothesizing" - in other words, inferring conclusions using basic physiologic/pathophysiologic principles. The thought process goes, "Chronically increased afterload (e.g, from systemic blood pressure) results in a larger regurgitant fraction, and this causes problems over time... therefore, if you Valsalva under load and make your blood pressure increase [acutely], it should do the same thing. Better be safe than sorry and never lift weights."
Interestingly, this is the exact same reasoning used to argue against resistance training for patients with heart failure, or even the preposterous argument that resistance training thickens your ventricle to the point of causing heart failure... even though we now have plenty of evidence showing resistance training to be 1) extremely beneficial and 2) not harmful in these patients. We similarly used this physiologic hypothesizing in the past to argue that giving beta blockers to patients with heart failure would be extremely harmful... turns out, that's not the case either, and now these medications are are standard therapy for chronic heart failure.
The point being, inferring conclusions about the potential harms of an acute exercise stressor based on chronic pathophysiology has failed us numerous times. It also tends to significantly overestimate the risk of harm, while completely disregarding the known benefit.
But in the absence of data, it's not surprising that people who 1) have no experience with training, and assume you're walking into the gym on day 1 to take 405 lbs out of the rack, and 2) have liability concerns, tend to err on the side of caution. Hell, even with data in numerous disease states they say the same things. We had two questions this week about people who said their doctors recommended against strength training - one individual with PCOS "because it will raise testosterone" and one with Crohn's disease "because it will cause digestive stress"... both of which are 100%, complete, utter bullshit (-- had to borrow Sully's bold/italics combo to convey my frustration on the matter).
However, when it comes to your situation it's still relevant to know what you mean by "relatively minor leakage" of the aortic valve (how much?), and why you have it (primary valvular issue vs. dilation of the whole aortic root?). We see "trace regurgitation" of one or more valves on just about everyone's echocardiogram, and this is of essentially no real significance. But if you had severe chronic aortic insufficiency with "wide-open" valves and a massive regurgitant fraction things get a bit hairier as you'd be considered for surgery at that point. This doesn't sound like the case, given your description and lack of symptoms or heart failure (though Carvedilol is an unusual choice for Atrial fibrillation).
We have no data to suggest that our style of strength training will cause "trace" AI to progress to severe AI. That concern is entirely based on the "physiologic hypothesis" and acute measurements of regurgitant fraction during exercise, not long-term trials following patients over time. Some people's AI progresses, other's doesn't, and I'm quite sure that if you happened to be strength training and the degree of leak progressed, your Cardiologist would be quick to blame the training. But we just don't know. If the leakage is in fact minor (what we would call "trace regurgitation") I would personally be comfortable training with a belt, because I perceive the benefits to outweigh the risks (admittedly, based on experience and without objective data). Beyond that, things get a bit more complicated and I cannot provide any recommendations on the matter.
P.S. Having "small pre-beats" is doesn't sound like a typical description of atrial fibrillation, but potentially more like PACs or PVCs... so I'm not sure of your actual diagnosis there. Are you on anticoagulation?
P.P.S Standard disclaimer, as Sully said. I am not your doctor, this is for educational purposes, it is not medical advice for anyone or anything, and you should always and forever obey your Cardiologist in all facets of your life.