Jesus of Nazareth was ONE PERSON -- fully God and fully sinless Man at one and the same time. So you cannot split Him as you have done.
Temptations are EXTERNAL. Yielding to temptation is INTERNAL. Therefore the temptations of Christ by Satan were all very real and totally external. But since Christ did not have an indwelling sin nature (being sinless Man) He could not, and would not, and did not, yield to any temptations. And that is exactly what Scripture says -- "without sin" or apart from from sin.
Then you have God being tempted...because Jesus is God.
But if the only temptable part of Jesus was His humanity, then God was not tempted; but it was only the human part of Jesus.
Now in the hypostatic union Jesus is 100% God and 100% Man...but I do not see anything in that doctrine that says that the human part of Jesus and the divine part of Jesus are absolutely one, so that if He is tempted in His humanity He must also be tempted in His Deity.
But if it be the case, I would say that the Lord God can be tempted...for it is written so in Matthew 4:7...but I would say that it is only possible inasmuch as the Lord God has taken on human flesh...therefore the Son can be tempted in His humanity; and if it is also necessarily in His Deity then it was also possible for God to sin, and we are back to square one.
But one thing we know....that Jesus Christ before He descended...the Father who inhabiteth eternity...it is impossible for Him to sin and He cannot be tempted with evil.
Now God cannot change...Malachi 3:6, Hebrews 13:8...and therefore when He took on flesh, the Spirit of Him did not suddenly become able to sin or be tempted...however He did take on a new nature of humanity, and this humanity was able to be tempted and to sin.
I think that we must all agree that Jesus Christ is the holy God...yet He was tempted in all things just as we are, and yet without sin. I consider that in His humanity He was temptable; but that in His Deity He was safeguarded against succumbing to the temptations presented to Him. When He was hungry after 40 days of fasting, He was tempted in His humanity...the God-aspect of Him was not tempted but He was tempted in His flesh...but the God-aspect of Him would have made Him immune to actually giving in to something that would have compromised the unchangeable character of God within Him. To the extent that His divine nature would have been compromised, His divine aspect could not even consider succumbing to what was laid before Him...His consideration of doing what would have changed His immutable nature (in His Deity) was entirely in the humanity of His person.