We normally assume that a reduced thrust take off (not runway / obstacle limited) considers the climb performance and aircraft handling, thus there should not be any requirement to increase thrust, but if the crew elect to do so then there should not be any handling problems.
However, if a reduce thrust take off is chosen specifically to reduce V1 (runway length limited) by reducing Vmcg, then there might be reason to reconsider any other effects if full thrust were to be applied after an engine failure. If V2 is also rescheduled to a lower value, as a function of a reduced V1, then the Vmca considerations might also have to be reconsidered. (JAR/FAR 25.125 / 25.149).
Thus as speculation/debate, any restriction on increasing thrust or not, might be taking the latter point into consideration and some of the reasons for this are in the definitions of ‘Reduced Thrust’ and ‘Derate’.
AC 25-13 'Reduced and Derated Takeoff Thrust Procedure' (note date – 1988) and AC120-62 appear to support the observation above:
4. b Derated takeoff thrust, for an airplane, is a takeoff thrust level less than the maximum takeoff thrust, for which exists in the AFM a set of separate and independent, or clearly distinguishable, takeoff limitations and performance data that complies with all the takeoff requirements of Par…. When operating with a derated takeoff thrust, the value of the thrust setting parameter which establishes thrust for takeoff is presented in the AFM and is considered a normal takeoff operating limit.
c. Reduced takeoff thrust, for an airplane, is a takeoff thrust less than the takeoff (or derated takeoff) thrust. The airplane takeoff performance and thrust setting are established by approved simple methods, such as adjustments, or by corrections to the takeoff or derated takeoff thrust setting and performance. When operating with a reduced takeoff thrust, the thrust setting parameter which establishes thrust for takeoff is not considered a takeoff operating limit.
This definition is expanded in AC120-62 “ … the thrust may be restored to the maximum level (or derate) as appropriate for the conditions of the flight at any time during the takeoff. i.e reduced thrust can be lower than a fixed derate
5. b. Relevant speeds (VEF, VMC, V1, VR, and V2) used for reduced thrust take offs are not less than those which will comply with the required airworthiness controllability criteria when using the takeoff thrust (or derated takeoff thrust, if such is the performance basis) for the ambient conditions, including the effects of an Automatic Takeoff Thrust Control System (ATTCS), It should be noted, however, that in determining the takeoff weight limits, credit should not be given for an operable ATTCS.
AC 25-7A Flight Test Guide for Certification of Transport Category Airplanes, also appears to supports this view (Para 91 and Appx 4).
Thus aircraft using an assumed temp reduced thrust may have no restriction on increasing thrust, but derated operations might have; is this the difference seen in the manuals?
Link to FAA ACs / search