Indeed, if the referent of the 1 Tim 6:16 is Jesus, the understanding that Jesus is the only one who is eternally and intrinsically indestructible, just like the Father, is the most plausible interpretation, because otherwise, had He been a creature and not eternal, then the phrase of Paul would lose the sense, for if Father just gives immortality to Jesus as to a creature, so that Jesus has immortality by participation and not intrinsically and inseparably (and, thus, eternally) then there will be no ontological difference between Jesus and any holy man, say a prophet Daniel, for both would have immortality bestowed upon them by the Father, who would be thus the only one properly speaking immortal. However, Jesus is ontologically different from all, be they greatest angels or greatest prophets and saints. Paul stresses this elsewhere as well, saying that when even apostles (the most ardent aspirants towards God's Kingdom) fall or falter, "He remains steadfast, for He cannot deny Himself" (2 Tim. 2:11-13), just like to say "if everything will dry out, water (in all its conditions) will remain wet, for it cannot deny itself", (and indeed, on the condition that water remains, it will be wet necessarily), for God cannot not be God.
Paul speaks, in fact, by this expression, of Jesus properly possessing the immortality, that without Him nobody can have immortality. Father ontologically cannot bypass Jesus when He bestows immortality to humans, for Father cannot give immortality unless through Jesus, who shares proper immortality with Him for all eternity, not any less than, to give an analogy, a sun cannot enlighten without its rays, for as rays and radiation is intrinsic to the sun, so also - yet in an incomparably sublime, indivisible and unapproachable way - the Son/Logos is intrinsic to the Father. What is greater, to create and bring from non-existence to existence, or to keep something in existence after the creation? According to the brilliant insight of Descartes, absolutely the same exertion of energy is needed both for creation (i.e. bringing something from non-existence to existence) and for the keeping of this creation into existence. If so, then as the Father is completely and perfectly impotent to create world without His Logos, so also, He is similarly impotent to bestow immortality on anybody without His Logos, without Jesus, who is the very Logos incarnate.
Just for a little fun: can one imagine such a situation: Father wants to give immortality to the late Nelson Mandela, and usually He does so through Jesus, but this time He decided not to bother Jesus and do it immediately by His own separate agency; Jesus sees then Nelson Mandela immortalized and asks: 'Father, I have no reason to object, but still why did you do it without me?', 'Oh, My Only-Begotten, don't be cross on Me! You were fatigued, so I decided to spare You'" and such kind of absurdity; a gross absurdity indeed, because the Father's action is impossible without the immediate co-action of the Son. Exactly that is what Jesus says in John 5:30 "By Myself I can do nothing", and more clearly in John 5:19: Jesus gave them this answer: "Very truly I tell you, the Son can do nothing by himself; he can do only what he sees his Father doing, because whatever the Father does the Son also does". This statement necessarily implies that neither Father does anything without the Son, for if "whatever (i.e. everything that) Father does the Son also does" then all actions of the Father, among them the action of creation and that of immortalisation is also done by the Son, while the Son must be uncreated, because had Father acted the creation of the Son as well, then given that He (Father) does only those things that the Son also does, there will obtain an absurdity that the Father acts the creation of the Son with the Son co-acting. And Paul's theology is in no manner different from that of John in this respect.
However, one still can say that Father "gives" immortality to Logos in a sense that Father is the eternal Source of the Son, so that what the Son has, He has from His eternal Source, but without any temporal succession, for this is instantaneous and eternal and essential giving, for the Father gives everything to the Son, and everything means the entirety of His essence/being, so that the Son's being is the very same being as the Father's.
Grammatically, the subject of the 1 Timothy 6:16 can be both the Father and the Son, and theologically there is no problem with any of the option, given the arguments presented above; but perhaps, stylistically Father is more fitting. The notion that this subject "nobody can see" does not exclude Jesus as a referent, even though Jesus was seen in His earthly sojourn, because Paul distinguishes the physically tangible Christ from the invisibly and innately perceived Christ (2 Cor 5:16), but when we perceive Christ in this latter way, that is to say, through the "Mind/Spirit of Christ" indwelling in us (1 Cor 2:16), then we also perceive the Father, for Jesus Himself claims that Father is "seen" in this way (cf. John 14:8-10). Thus, in Their divinity neither Father nor the Son are seen or can be seen either by bodily eyes or by the best of human intellectual efforts, but Both are "seen" through the gift of Spirit.