Probably, the method seems pretty straightforward so long as you think ethics are for suckers. Let's start by touching on some of the points you bring up. Forcing someone else's gate open is not a well understood phenomenon and probably doesn't serve a meaningful purpose in this case. Forcing someone else to open their gate of truth is possible but is far from trivial, it was only done once by Pride to Mustang and given how fervently the homunculi tried to get Mustang to perform a human transmutation of his own volition before Pride forced the issue, how Pride notes that it took a lot out of him and we see his philosopher's stone fueled body crack slightly afterwards we can surmise it was a transmutation that leveraged the power of the philosopher's stone as an alchemical catalyst and cost a significant amount of the philosopher's stone's power that Pride would not have used if he hadn't been forced. With that in mind it doesn't seem likely a human without a philosopher's stone to serve as a catalyst could force someone else's gate open. Why not just draw the human transmutation circle for the child and have them activate it willingly, no philosopher's stone needed.
Truth always exacts a toll from those who attempt human transmutation and sends them through the gate where they acquire vast alchemical knowledge. We don't know if Truth accepts philosopher's stones, gates, debit, etc. as payment to waive the usual toll, it's never come up. But for the task at hand why not just pay the toll? Truth doesn't tend to exact enough of a toll to kill you, with the exception of Alphonse (even if your kid gets Alphonsed you could just pull an Edward and bind their soul to something while you try to revert the situation). Then when they get back missing an arm, their hearing or whatever just do what Ed, Al, Mustang and Izumi refused to do and use a philosopher's stone to regrow/fix whatever was taken like the homunculi do. At the start of FMA Edward and Alphonse knew once they had gotten their hands on the philosopher's stone they could transmute their broken bodies whole again but once they learned about the true nature of the stones they adamantly refused to use the lives of others to fix their mistakes. If you have no such qualms you're good to go.
If you have some qualms and are looking for a less morally bankrupt way to pay for a limb there are possible replacements. Edward demonstrates you can exchange your gate of truth as part of an alchemical exchange but it is made clear such an exchange is not meaningful for others. The gate of truth is the only way out of Truth's domain, without it there is no way to return to reality. Edward is a special case because his gate is somehow linked to Alphonse's gate, when Edward gave up his own gate he and Alphonse left through Alphonse's gate together. Mustang later confirms transmuting one's gate is useless for most when he expresses he is down to swap his gate for his sight but laments not having a way back afterwards. The fact that Edward and Alphonse share a connection through their gates is an unnatural occurrence and the mechanism and reason for it isn't well established, all we know is the linking of the gates probably happened the day the boys tried to transmute their mom (whether it be from the human transmutation itself, Edward swapping an arm for Al's soul or Edward binding Al's soul to an object) since it was later established Alphonse's body trapped in Truth's domain derives nourishment from Edward through the connection of their gates which necessarily began here. It's unclear if you can restore the body parts of someone else without your gates being linked and you certainly will end up trapped in Truth's domain if you don't have a linked gate to exit from.
Rather than the lives of others could you exchange your own life for a limb? You can in fact exchange life force in a transmutation, philosopher's stones are the condensed form of the life forces of many people, their use as an alchemical catalyst is shown repeatedly throughout the series, from regenerating the bodies of the homunculi to allowing Father Cornello to seemingly subvert the law of equivalent exchange all the way to possibly allowing Pride to force Mustang's gate open. The philosopher's stone however does not subvert the law of equivalent exchange, it only balances out otherwise unbalanced exchanges with life force. The philosopher's stone is not limitless in use either, eventually all of the life force contained within is expended and it ceases to be, such as when Mustang fried Lust repeatedly until her stone ran out of power. On a more local scale one can use their own life force while alive as an alchemical catalyst which Edward demonstrated after getting impaled by a steel beam fighting Kimblee. He notes that he needs to expend some of his own life force to boost the transmutation needed to close his wound and at some point also points out that he himself has 1 soul's worth of life force (where a philosopher's stone contains many). Is 1 soul of life force enough for a limb? No idea, either way this method is still dubious since sacrificing someone else's life to restore a limb is the exact circumstance we are trying to avoid.
