Neither Part 121 nor Part 135 require that a pilot demonstrate all precision approaches authorized on every checkride. During a proficiency check, a pilot in command must execute at least one normal ILS approach, and one with a simulated engine failure. The pilot must also execute a non-precison approach.
You refer to 121.441, which uses Appendix F of Part 121 as the guide, the relevant portion copied below. In no way does it state that every precision approach must be executed, for which an operator is authorized. Regardless, this does not define the nature of a precision approach. The definition of a precision approach is specified in the opening part to 14 CFR.
A precision approach is clearly defined as one having vertical guidance, while a non precision approach is clearly defined as one that does not have vertical guidance, as previously given. How much more clear can this be?
Part 121, Appendix F, subparagraph (c):
(c) ILS and other instrument approaches. There must be the following:
(1) At least one normal ILS approach.
(2) At least one manually controlled ILS approach with a simulated failure of one powerplant. The simulated failure should occur before initiating the final approach course and must continue to touchdown or through the missed approach procedure.
(3) At least one nonprecision approach procedure that is representative of the nonprecision approach procedures that the certificate holder is likely to use.
(4) Demonstration of at least one nonprecision approach procedure on a letdown aid other than the approach procedure performed under subparagraph (3) of this paragraph that the certificate holder is approved to use. If performed in a training device, the procedures must be observed by a check pilot or an approved instructor.
Each instrument approach must be performed according to any procedures and limitations approved for the approach facility used. The instrument approach begins when the airplane is over the initial approach fix for the approach procedure being used (or turned over to the final approach controller in the case of CA approach) and ends when the airplane touches down on the runway or when transition to a missed approach configuration is completed. Instrument conditions need not be simulated below 100 feet above touchdown zone elevation.
Regarding 135.297, it would appear you alluded to subparagraph (b), which states:
"(b) No pilot may use any type of precision instrument approach procedure under IFR unless, since the beginning of the 6th calendar month before that use, the pilot satisfactorily demonstrated that type of approach procedure. No pilot may use any type of nonprecision approach procedure under IFR unless, since the beginning of the 6th calendar month before that use, the pilot has satisfactorily demonstrated either that type of approach procedure or any other two different types of nonprecision approach procedures. The instrument approach procedure or procedures must include at least one straight-in approach, one circling approach, and one missed approach. Each type of approach procedure demonstrated must be conducted to published minimums for that procedure."
This is not to be construed as meaning that a pilot must demonstrate every approach authorized that operator. However, if a pilot wishes to execute a precision proceedure that is authorized to that operator, then he or she must have demonstrated it within the preceeding six callendar months.
As we discussed by PM, your ops specs do not place an LDA with a glide sloope under the heading of non-precison, as this was changed two years ago to a heading describing approaches other than an ILS, MLS, GLS, etc. This is an important distinction, and it was made with the intent of recognizing the upcoming of other types of precision approaches, or approaches which are not nonprecision.