How then does the biofilter screen out contaminants? If the matter is not pulled apart in the matter stream, how can the filter screen out and remove the foreign molecules?
That shouldn't be a bottleneck. Let's say the matter is not destroyed and then recreated, E=mcc style - it's merely converted, and thus retains structural information about itself. But in both the converted and unconverted state, the sensors can sense objects as coarse as bacteria and virii, and try and eliminate them the best they can. That's an exercise orders of magnitude easier than trying to memorize the positions of all the atoms, let alone smaller particles or other states of existence, of the transportee body. It doesn't require "quantum level scanning" or "total conversion from matter to energy" - something as coarse as bulk insertion of antibodies, or precision radiation treatment, would do.
The only way he could have created a second beam was to add matter (or energy to turn into matter) from somewhere to supply the needs of the second pattern.
Perhaps so. But the episode gives no suggestion that this additional matter or energy could have turned into a person without at least the temporary presence of the original "pattern", whatever that is.
In "Lonely Among Us", Picard's "physical pattern" is said to remain in the transporter when he performs a nonregulation "energy only" beamout. By latter terminology, that would be the phased matter stream in there.
So quite possibly, a beamout is a hybrid of a material part that carries most of the information, too (the phased matter stream) and an abstract "memo" that carries the rest of the information. The latter is compact enough to be stored in computers (like in "Lonely Among Us" or, say, "Our Man Bashir") but isn't enough to reassemble the person out of raw materials; either the former has to be used as material in order to complete the process, or then the former has to be used as an additional source of information in order to arrange the raw materials into the person.
That sort of an arrangement would be restrictive enough to prevent the use of the transporter as an infinite-resolution or even high resolution xerox machine, while allowing for infinite-resolution movement of items. A house of cards, perhaps, but one that takes the shape required by the episodes.
In any case, we cannot rely on "writer intent" here, because the writers never thought about such things, not to any meaningful detail anyway. What we need is a descriptive theory that fits the currently known facts and fills the holes with suitable putty. It's not very scientific, because it cannot be tested in any manner - but if it works within its region of applicability (that is, all known episodes and movies), then it's good enough.
Except that when they did cure her, she did seem to remember everything.
Not really. When she stepped out of the transporter, all she did was stare, and get hugged by Picard. She didn't get a word out of her mouth.
We then cut to a completely different scene, where apparently Picard has brought her up to speed, and the two are discussing the operation. There is no discussion of the events of the past days or hours as such.
For all we know, Pulaski lost everything that happened between now and the last time she brushed her hair.
In sharp contrast, in "Rascals", Picard immediately reacts after the transporter releases him: he tries to stroke his now-missing hair, and cracks a joke (?) about everything looking smaller now.