The physics of twisted bilayer graphene away from the exactly solvable chiral limit and the quantum Monte Carlo sign-problem-free charge neutrality point is elusive due to the exponential increase in the computational complexity, which has rendered explanations of experimentally observed insulating and superconducting phases restricted largely to the perturbative level. Here we focus on the filling factor ν=±2 and address the question of the strain dependence of the interacting ground state by approximate quantum Monte Carlo (AQMC), state-of-the-art exact diagonalization (ED) and Hartree-Fock (HF) mean field, in order to investigate the strain-tuned transition from the Kramers intervalley coherent (KIVC) state to the incommensurate Kekulé spiral (IKS) state. While all three methods capture the KIVC order, only ED and HF detect the weaker IKS order that AQMC does not capture adequately. As the AQMC is still capable of capturing stronger orders like the KIVC, our combined protocol may open the door for further understanding of the rich phases of twisted bilayer graphene and other strongly-correlated systems.