I'm not sure, but there may be a confusion happening. You are taking an LGPL library (let us call it "L") and combining it with other libraries, themselves under BSD and MIT licences (let us call these libraries "B" and "M" respectively), to produce a new library (let us call it "Z"). In addition to combining itL with B and M, you may be making modifications to L during this process.
When s5 refers to "the Library" (note the capital, which is significant) it is referring to L. When it says that you must "accompany the combined library [that is, Z] with a copy of the same work based on the Library, uncombined with any other library facilities, conveyed under the terms of this License." it means you when you give someone a binary copy of Z, you must also provide the source of L, with any modifications you have made to it during this exercise, unencumbered by the source of B and M, and you must convey this source under LGPL.
You do not have to provide the source of B and M, even though they are freely-licensed (though of course you may if you choose). You will also have BSD, MIT, and LGPL s5b obligations with respect to the distribution of Z, which I do not discuss here.
As Bart notes, if you have actually edited B and M into the source of L, rather than just combining (possibly modified versions of) B, M, and L into a single library Z, this analysis doesn't apply.