Peak power scaling of semiconductor disk lasers is important for many applications, but their complex pulse formation mechanism requires a rigorous pulse characterization to confirm stable fundamental modelocking. Here we fully confirm sub-300-fs operation of Modelocked Integrated eXternal-cavity Surface Emitting Lasers (MIXSELs) with record high peak power at gigahertz pulse repetition rates. A strain-compensated InGaAs quantum well gain section enables an emission wavelength in the range of Yb-doped amplifiers at ≈1030 nm. We demonstrate the shortest pulses from a MIXSEL with a duration of 253 fs with 240 W of peak power, the highest peak power generated from any MIXSEL to date. This peak power performance is comparable to conventional SESAM-modelocked VECSELs for the first time. At a 10-GHz pulse repetition rate we still obtained 279-fs pulses with 310 mW of average output power, which is currently the highest output power of any femtosecond MIXSEL. Continuous tuning of the pulse repetition rate has been demonstrated with sub-400-fs pulse durations and >225 mW of average output power between 2.9 and 3.4 GHz. The strain-compensated MIXSEL chip allowed for more detailed parameter studies with regards to different heat sink temperatures, pump power, and epitaxial homogeneity of the MIXSEL chip for the first time. We discuss in detail, how the critical temperature balance between quantum well gain and quantum well absorber, the partially saturated absorber and a limited epitaxial growth quality influence the overall device efficiency.