Leaning aluminum will be tricky on .040" because you need to manipulate small distances and motions, like a doctor performing a surgery. Leaning on .125" thick aluminum is I think a lot easier for most people, who are not surgeons.
I would go with the .040" tungsten, as long as the tip is not "melting / balling", and try and make your joint fit-up as tight as possible, if you get it close enough you might even be able to get the two parent metals to "fuse" together for a tack weld, without adding any filler. They need to be not only clean (of aluminum oxides), but touching each other right in the spot you are attempting to tack weld, for that to work with aluminum.
Also, are you conscious of your torch angle, and your arc length? Keeping the arc length as small as possible will keep the heat focused more. Also, I would probably aim the torch (torch angle) into the larger flat surface that is not melting back, to direct the heat towards it. Ideally you want a puddle to form on both sides at the same time, in fact it is critical if the pieces aren't touching, and then you'll further have to jab the filler rod right into the middle of the two molten puddles at the proper time, to let the puddles flow together as one. But welding in the middle of a surface needs more heat input than welding on the edge (as with a lap joint) for both to get to melting temperature at the same time. You can direct the heat to one piece or the other by controlling your torch position and torch angle.
Also there are some machine settings you may be able to try - depending on what machine you are using. Using more DCEN% (balance) will also focus the heat input closer to the tip of the tungsten. As will increased AC frequency, or some high speed pulsing. (But if your machine can't control all that, don't worry, there are still plenty of other variables you can play with to hopefully create the desired effect.)
I have trouble dialing my machine in for very thin materials needing very small welding currents, because near the low end of travel of my "current" knob, I find a small adjustment of the knob makes proportionally a very large change in the current setting. If I turn it all the way minimum, it tries to weld with just 1 amp, and will not be able to establish a normal welding arc (won't get past high frequency starting mode.) Then with a knob adjustment, for some reason it likes to try and jump up to 6 amps (a 600% increase.) etc. Wondering if something like that might be causing you problems on the thin material too.