The Ark is still buried and hidden. Like Moses' body it is not going to be found. Even if it were found (which it will not be) it has no power in it, I mean, DUH, because God is no longer indwelling in it. Like the tent that the Israelites pitched so that God's presence in the cloud could be seen as being present among the Israelites, the Ark is the same thing: only a place to increase the people's faith by SEEING and thus believing that God is with them. Once Israel started to fall and was afflicted, with God's permission, by enemies, the Ark was hidden and of course God no longer indwelled in it, duh.
The ark in Ethiopia is a copy. Stop the ridiculous fantasizing that any King of Israel would give the ark to some woman he had nookie with.
Remember, Solomon's life was CUT SHORT by God as punishment for building worship places for idols to please his pagan wives and concubines. You have to stick with what is in the Bible, friends (and the Qur'an). No where does God permit the Israelites to do something as ridiculous as give away anything from the Temple, certainly not his seat. Solomon was punished in his old age as he became a fool and collected women like baseball cards, and thus collected idols accordingly. His foolishness was cut short before it became any worse than that. But even in his greatest folly, and that of his descendants, no one would even think of giving the ark away. Rather, people would have died around it if it came to that.
No, as you know in the Bible holy things are hidden under siege. That is what was done. It was well within their skills to make one or two copies to fool occupiers.