First and foremost, whenever you bill anything that you know is wrong, you're violating state and government regulations. If an audit hits your facility, you'd likely find yourself in a place you don't want to be.
Second, if you practice this procedure over and over simply because someone is requiring you to do so in order to have your billing system calculate RVUs correctly, then you need to say "No." If there's an issue with the system not capturing data unless billing violations are knowing done, then it's the responsibility of the business to fix their system. That's just screaming all sorts of violation/illegal-ness up, down, and sideways.
Third, you say the clearinghouse removes the 51 and sends the claim along its way, but if a claim hits an edit, normally the claim would get kicked back for correction and resubmission. If the clearinghouse is indeed changing your claims and not telling you about it, then someone needs to address that practice because a clearinghouse should not be doing activities like that at all.
Fourth, by submitting a claim that's wrong just to calculate RVUs and relying on the clearinghouse to fix it and send it on, then essentially you've created a work-around to, well, basically cheat the system. If it were me, I'd turn and run the opposite direction as quick as possible.
Now, regarding your question, you cannot apply both a 50 and 51, it just wouldn't make sense even without the rule. Billing with a 50 indicates a price adjustment to 150% and a 51 indicates an adjustment down to 50%, so basically by adding a 50 and 51, you're asking for 150% and then asking a reduction of 50%, so in the end you get 100%. Directly from CMS: "Modifiers should never be added to claims unless the documentation has been reviewed and the use of the modifier is appropriate based on the documentation."
Bottom line here is that you're knowingly and actively submitting claims in which you know the coding is wrong. IMO, you need to stop throwing yourself under the bus (because the finger WILL get pointed at you) and refuse to do it any longer. If it means the RVUs are off, then that's not your problem. By refusing to submit incorrect claims, coding and submitting them properly, then you're doing your job correctly. Period. It's up to the facility to fix their reporting system to capture the RVUs and purposely coding incorrectly is NOT the answer.