A number of points in the NT and Phil. 1:19 make me think that Paul is speaking of the holy spirit when he talks about "the spirit of Jesus Christ."
(1) Paul says that both the prayers of the brothers and sisters as well as the spirit of Jesus will "result in his deliverance" (Emphatic Diaglott) or his "salvation" (NWT). Scholars are not sure whether the SWTHRIAN mentioned refers to eternal salvation, deliverance from prison, or vindication in a legal sense. But regardless of what "salvation" Paul is talking about, he most certainly has in mind his eternal destiny as well as a possible release from prison (this may be an example of deliberate ambiguity). But how would this "release" come about? Would it happen through the mental disposition of Christ manifested by Paul or through the holy spirit that God vouchsafed to Christ? In answer to this question, notice that Paul associates the spirit of Jesus with the prayers of the first century brothers and sisters in Philippi(Cf. Acts 4:23-31).
But why didn't Paul call the "spirit of Jesus Christ" God's spirit if they are in fact one and the same? Well, it is interesting that Paul entreated the Lord three times, begging God to remove a thorn that apparently plagued Paul for quite some time (2 Cor. 12:8). What was the result of this prayer? Jehovah told Paul that his power was perfected in Paul's weakness. Consequently, the Apostle said that he would boast in his weakness, "so that the POWER of the Anointed" would abide upon him (2 Cor. 12:9 Emphatic Diaglott).
Notice that DUNAMIS is first described as God's power, then it is called "the POWER of the Anointed" (Christ). But how would Paul be infused with the Anointed One's power? Acts 1:8; 10:38; Eph. 3:16ff all indicate that the power of God is communicated via His holy spirit. I therefore conclude Paul believed that God and Christ work so closely together as they imbue believers with the holy spirit that to desire the spirit of Jesus Christ is to desire the spirit of God.
Furthermore, it seems that Paul most certainly refers to God's spirit as the spirit of Christ in Rom. 8:9-11. Those verses dovetail with Titus 3:5-6.
But notice closely what Gal. 4:6-7 also teaches us. It is not God's sons who cry out in 4:6, but the spirit of God's Son (cf. Romans 8:14-17). And note that it cries out, "Abba, Father!" I would submit that the spirit of 4:6 is thus the holy spirit that engenders a conviction of sonship in God's children. Thus, while all Christians have (or are supposed to have) the dominant mental attitude of Christ, not all Christians have the conviction of sonship. So that makes me wonder: since the spirit in 4:6 cries "Abba," how could it refer to Christ's mental disposition? Why would Christ's mental disposition cry Abba in a worshiper of Jehovah? Surely it is not Christ's "mind" that produces God's sons, is it?