Then explain how David could have "believes on the Lord Jesus Christ" in scripture.
And why would Peter state 1 Peter 1:6-9 to the nation of Israel?
Well David certainly did.
“«A Psalm of David.» The LORD said to my Lord, "Sit at My right hand, Till I make Your enemies Your footstool."” (Ps 110:1 NKJV)
Jesus explains that in that verse, David was referring to Him, to Christ:
“Then Jesus answered and said, while He taught in the temple, "How is it that the scribes say that the Christ is the Son of David? "For David himself said by the Holy Spirit: ‘The LORD said to my Lord, "Sit at My right hand, Till I make Your enemies Your footstool."’ "Therefore David himself calls Him ‘Lord’; how is He then his Son?" And the common people heard Him gladly.” (Mr 12:35-37 NKJV)
Peter wasn't writing his first letter to the nation of Israel. He is writing to Christians who have been dispersed by persecution:
“Peter, an apostle of Jesus Christ, To the pilgrims of the Dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia, elect according to the foreknowledge of God the Father, in sanctification of the Spirit, for obedience and sprinkling of the blood of Jesus Christ: Grace to you and peace be multiplied.” (1Pe 1:1-2 NKJV)
It is to Christians that he writes in vv 6-9:
“In this you greatly rejoice, though now for a little while, if need be, you have been grieved by various trials, that the genuineness of your faith, [being] much more precious than gold that perishes, though it is tested by fire, may be found to praise, honor, and glory at the revelation of Jesus Christ, whom having not seen you love. Though now you do not see Him, yet believing, you rejoice with joy inexpressible and full of glory, receiving the end of your faith — the salvation of your souls.” (1Pe 1:6-9 NKJV)