My first greyhound had heart disease, and like the others have said, did not need anesthesia for her echocardiogram. Maybe since Daisy is a spook, just a little Xanax or something like that to calm her would be enough. If not, I would agree with saying no. Sugar started on lasix and enalipril, then moved on to vetmedin as she got worse. All of this lasted about 3 years, then I lost her to kidney failure.
The grain free food/heart disease issue is fairly new over here. Some tests have shown a possible link, and since most dogs don’t really need grain free it seems sensible not to go there. The issue seems to be (this is all still very early in testing) not so much the lack of grain as the often exotic things they replace the grain with. Sugar is the only dog I have ever had with heart disease, but I got her at age 7 from a not so great situation, so she probably didn’t get good care earlier in her life. All my dogs have been mostly raw fed.