Aircraft Technician reported improper procedure used to perform aircraft maintenance.

Narrative

On Day 0 at XC50; Aircraft X went AOG at APA for WOW MISCOMPARE Amber CAS: 'After Takeoff landing gear would not retract. After several attempts to retract gear WOW MISCOMPARE Amber CAS message was generated'. At XD32; I requested the aircraft to be placed into hanger to thaw. I informed Maintenance Control that we would not be able to do gear swings with absence of a hydraulic mule. At XD39; Person A in Maintenance Control responded that we could do with the aux pump if needed and attached Cessna letter XXXXXXXX. At XJ16; I sent a message informing Maintenance Control that the aircraft finally made it into the hanger and that we still need a Maintenance Manual (MM) reference for performing the gear swings with the aux pump. At XJ29; Person C at Maintenance Control replied that he would look into finding a reference; and if he could not find one that he would inquire Person B for support documentation. At XK35; Person D sent an email expressing his concern for an approved procedure that allows us to deviate from what is written in the MM. At XK53; Person C sent an email stating that they were not looking to have all the operational gear swing checks complied with but just wanted to make sure the gear will go up and locked and come back down and locked. He added that if we weren't comfortable with doing this that they could pursue a special ferry permit. At XK57; Person D replied; please pursue working on a Special Ferry Permit. At XL32; Person C responded that they were going to look for another vendor and would not prefer to ferry due to an improper preflight. At XM03; Maintenance Control notified us that they were going to seek a vendor as an assist for a hydraulic mule and don't close out our paperwork. On Day 1; XA49; we received an email from Person E stating that after speaking with Person F and Maintenance Control; that they felt that the letter from Cessna was acceptable for use to continue the directed maintenance. At XA50; Person E called us to follow up on his email and stated to get the aircraft back to the customer. At XB16; Person D and I spoke with Person F and he in turn sent us an email granting his authority to use the aircraft aux pump to perform a gear swing to assure proper gear indications. We went ahead and complied with the maintenance as directed.Maintenance Control directed us to perform gear swings per a correspondence letter XXXXXXXX dated 2 years ago from Cessna that references servicing of the hydraulic system per chapter 29. No where in the letter does it explicitly state that the aux pump can be used to swing the gear. There is no maintenance manual reference that states that you can use the aux pump to swing the gear. On Day 1 at XD17; spoke with Person G with Cessna and he informed Person D and I that gear swings with the aux pump is not prohibited but is not recommended. Person D and I did not feel comfortable with deviating from the maintenance manual and felt that the letter from Cessna lacked intent and was ambiguous. If any form of gear swing is prohibited with the hydraulic aux pump; the MM should state this or a more explicit letter from Cessna should be prepared. We were placed under utmost human factors of stress and duress from Maintenance Controls and our Supervisors to go against years of safety training and to speak up when put into a compromising position. Only after our phone call with Person F did we feel that we could execute the maintenance requested with his authority.