It is not healthier. To the contrary, it aggravates side effects and complicates recovery (due to further suppression of your endogenous testosterone).
You must understand that most of the guys pushing testosterone to everybody's cycles are not much brainy and they don't take steroids for athletic purposes. First of all, they want to become "huge" in order "to bang as many chicks as possible".
I have done 4 cycles with Anavar and it would never occur to me that I would have to take testosterone with it. My friend, who is a competetive athlete (hush!), takes the same Anavar, too, and he never complained about anything. To the contrary, he always feels great while on it.
O.K., the addition of testosterone will make the cycle more effective, no doubt, but since many people take Anavar to avoid side effects associated with steroid use, I don't see much sense on it. If you don't care about side effects, you can take Winstrol instead, and the cycle will be both more effective and less costly.
But if you ask, how much testosterone you must inject to counterbalance suppression, count with me: Your body produces about 7 mg testosterone/day (50 mg/week). Since the ester of testosterone enanthate or cypionate makes up about 30% of the total weight of the steroid, you should take at least 70 mg/week. (Or 65 mg testosterone propionate/week). In reality, this is somewhat simplified, because the esters have different half-lives, but apparently, you don't have to take more than ca. 100 mg testosterone ester/week.
Alternatively, you can inject HCG throughout the cycle to prevent your hormonal system from suppression. If you are not extra sensitive to androgens, the endogenous testosterone running in your body shouldn't much elevate the total androgenic activity.
If your primary aim is the improvement of strength performance, then you don't have to care about body fat, of course. But if you expect that Anavar would improve your body appearance, then I agree with your advisors that losing some fat would be certainly beneficial. You can't namely expect that you will gain much muscle on Anavar. This steroid primarily produces strength and pure myofibrilar hypertrophy (pure muscle) without water, which leads to the effect of "hardening" and "cutting" of your muscles. If you don't lose fat, you won't see much change on the shape of your muscles and you will be disappointed.