The priest certainly has the final say on the readings.
While many priests and parishes have lists of readings, there is no generally required list. That is, the Church itself does not have a required list from which you must chose. That being said, there are many good reasons why a priest or parish may have lists from which they allow you to choose: it gives you some guidance and assistance to choosing appropriate pre-approved readings; it gives you a way to participate in planning; it makes things simpler for the priest in getting his homily together; it makes sure the message is coherent; etc., etc. Most parishes have easily accessible guidance on their websites.
If you want something different, I would suggest speaking with the priest. He may well be willing to accommodate you. On the other hand, he may not - again the choice is his. I think I have never known a priest not to be approachable about this matter, but like any human being he could have a bad day - or, if you are planning way ahead, he may put you off as he prefers to deal with more pressing matters.
Pre-planning at the funeral home can be a good idea. I personally think speaking with family, friends and your own priest is the best - especially if disease or other conditions indicate the time is approaching. If you simply are getting things in order and there is not an expected need any time soon, it is a bit “iffyier.” Funeral homes can be forgotten about or close; you might move; priests retire, die, or move; you might change your mind,… The best thing in this case is to make sure your family knows now - what you want and where you keep instructions. Talk with them on a regular basis (but not to the point of them tuning you out!)
Personally, I’ve always liked the profundity and beauty of the beginning of the Gospel of John, and thought it would make a great basis for a funeral sermon. Yet, it is not specifically about death so it never seems to make the cut on any lists.