This is always a weird question in my mind. How can you possibly choose God without faith? And if God gives you faith to choose God, then won't you choose Him because you have faith?
Essentially what you (and others) deny is that God gives you faith to choose Him.
Christ compared faith with a mustard seed. That was in response to a direct request to increase the faith of the apostles:
5 And the apostles said to the Lord: Increase our faith. 6 And the Lord said: If you had faith like to a grain of mustard seed, you might say to this mulberry tree, Be thou rooted up, and be thou transplanted into the sea: and it would obey you.(Luke 17)
What do we learn form this? Faith is not binary; it can be increased, or grown, and presumably lost (see 1 Timothy 1:19-20 for that). Indeed, many disciples chose to follow Christ but their faith was weak (see Luke 9:61-62, as well as the preceding episode described in Luke 9 and Matthew 8).
Christ continues in Luke 17:
7 But which of you having a servant ploughing, or feeding cattle, will say to him, when he is come from the field: Immediately go, sit down to meat: 8 And will not rather say to him: Make ready my supper, and gird thyself, and serve me, whilst I eat and drink, and afterwards thou shalt eat and drink? 9 Doth he thank that servant, for doing the things which he commanded him? 10 I think not. So you also, when you shall have done all these things that are commanded you, say: We are unprofitable servants; we have done that which we ought to do.
Faith, we understand now, is not only a source of good works of love but also a product of them. Grow in charity and your faith will grow:
Canon 24. If anyone says that the justice received is not preserved and also not increased before God through good works,[125] but that those works are merely the fruits and signs of justification obtained, but not the cause of its increase, let him be anathema.(Trent)
No, not if you read carefully. What I see being denied is that you have the ability to choose at all.
I don't know if this applies to you however, whether you follow the doctrine of, I think the term is, double predestination.