Don’t worry, so was I! I have done more research into this as a result of all your questions, and some of the information I provided was not accurate. My apologies for that. Here is a more up-to-date information. Firstly, when a child of record is blessed, apparently a membership record of some kind is prepared for the child, although I am not clear yet exactly what purpose that serves, because nobody can properly become a member of the Church without being baptized, which cannot happen until they are eight years of age. Maybe a membership record is prepared in anticipation of his joining the Church when he/she turns eight. I have to make further inquires abut that to find out more about it, which I cannot do right now. Secondly, it is expected that when a child of record turns eight, he/she is to be visited by his local church to encourage him/her to join the Church. This is something that ought to be explained to the parents of the child (if they are not members of the Church) before the child is blessed. So my apologies again for giving incorrect information.
No, it cannot be done that way. It can be done in the home; but it must be done under the supervision of the bishop, and the persons performing it must be judged by him to be worthy of the task, and the proper procedure for keeping a record of the event etc must be observed, otherwise the sacramental nature of the blessing is not preserved. Let me explain that anybody having the priesthood in our church can give an ordinary priesthood blessing to anyone else who has the faith to request it, and no records need necessarily be kept of that. That equally applies to a child at the request of its parents. But that is not the same thing as the sacrament of blessing and naming of children, which is what we are now talking aobut.
From an LDS point of view, you certainly can, and you would be very much encouraged to do so.
zerinus