I would not do this for exactly the reasons you've asked about:

1) By sharing the bathroom vent pipe, you're ensuring that at least half the gunk that's being spewed onto your porch is going to be spewed into your bathroom unless you can find and install some kind of one-way flapper between the new "Y" fitting and the bathroom. If you do find and install such a thing, it's going to be difficult to keep clean and operational.

2) I'm assuming since you say "first floor" that you also have a second floor, in which case that's a long way to go to get up to the roof. It's not such a big deal for the bathroom fan to vent up there, but that's a long way for lint from the dryer to be carried, and it's going to be more likely to collect on the walls of the pipe over that long distance.

Do you have a crawlspace or basement that you could run a new pipe through?

I cannot say I'm a fan of the design of that vent - although looking at your other question, I'm rather surprised that lint is falling down into that gap. Possibly your fan is creating too much pressure, and it's backing up inside the vent at the 180 and thus air is escaping through the pushing some lint down into that gap, where it can get stuck.

If that's the case, then you might try installing a baffle inside the vent to force lint up far enough in the vent so that it has to go out the intended port, but which still allows water out. Like so...

Sorry, the color doesn't show well, but inside I've added a piece of metal running from just below the intake port to nearly the end of the angled portion of the vent, with a brace connecting it to the body to keep it stable. You can't see it but there should be drip holes at the point where the new plate hits the flashing for water drainage.

The idea being, lint flies in and cannot be forced directly down to the gap in question, but water will hit the tray, run back and go through the drip holes.

That or replace it with this