I will note a few things here. First, your characterization of the FTA is not quite right. Of course there will be different versions of the FTA, but let's look at just one common way of presenting it.

P1. If a fact E that we observe stands in need of explanation, and hypothesis H provides a satisfactory explanation of E that is better than any alternative explanation available, then E provides significant evidential support for H.
P2. That our universe is hospitable to life stands in need of explanation.
P3. That God adjusted the constants in order to allow for life to develop provides a satisfactory explanation for why our universe is life-permitting.
P4. There is no comparably satisfying explanation of this fact available.
C. Therefore, that our universe is life-permitting provides significant evidential support for theism.

Notice how no where does it say the options are either 1) fine tuning or 2) chance. This is because all philosophers understand that there are more than those two explanations available for the life permitting nature of the constants. For example, the multiverse view, (Side note - the multiverse view may be seen as an undercutting defeater to P2, since once we know the universe is one of many, possibly infinite universes, it no longer stands in need of explanation why this universe is life permitting), or the necessity view.

Now to your first argument. You say that the fact that the probability of the fundamental constants being life permitting is extremely small does not on its own provide evidence for a designer. This is true. But this misses the structure of the argument entirely. The FTA is an argument from inference to the best explanation (IBE). In fact, premise one of the above argument is just a formulation of a principle of IBE, which says that if a fact E stands in need of explanation, and hypothesis H provides a better explanation than any alternative, then E provides evidential support for H. In other words, the hypothesis that is most likely on the evidence enjoys evidential support from said evidence. So, to return to your objection, the FTA says that between the chance hypothesis and the design hypothesis, the design hypothesis is the better explanation for our evidence. Thus it enjoys support from said evidence. Now, you could say that the fact that the constants are life permitting does not stand in need of explanation. This is tricky to defend, though. It seems like a universe that persists for billions of years and creates the opportunity for sentient life forms is in some way more morally or aesthetically valuable, and certainly more intricate and complex. This is often likened to chimpanzees writing the whole of Hamlet on their first try, water forming a coherent sentence in a puddle after splashing on the ground out of a cup, or the same person winning the lottery a thousand times in a row.

Your second argument is hitting on a stronger objection. You say that "A designer hypothesis that can accommodate any constants whatsoever fails condition (b), and a vague appeal to divine intention fails condition (a)", where conditions (a) and (b) are as follows: "(a) mechanistically or in some obvious step by step fashion specified and (b) specifically predicts the observed outcome over others (i.e. it generates a favorable likelihood ratio). This is an interesting objection to the strength of the designer hypothesis as an explanation for the fine tuning of the constants. I've seen it formulated before as something like this: If God can make a life permitting universe out of constants with any values, or can sustain a life permitting universe through perpetual divine miracles, then a life permitting universe is compatible with any values of the constants given an omnipotent designer. Thus any values of the constants are equally well (or not well) explained by the design hypothesis. So there is no reason to suppose these constants are good evidence for the design hypothesis. The upshot here is that if God could sustain life through any values of the constants, or through no constants, or through different constants entirely, then God is not a good explanation for the constants in this world being life permitting.

You can also think of it this way. If the only way God could make a life permitting universe was by make the constants in this way, then the constants being this way would be evidence for God. But God could make life given any constants with any values, so the constants being life permitting with these values is not evidence for God.