LoadArCoreSdk should check SDK version that is loaded, please see TODO found here: 
chrome/browser/android/vr/arcore_device/arcore_shim.cc:87

It might be sufficient to call ArCoreApk_requestInstall(), excerpt from
its documentation:
"If ARCore is not currently installed or the installed version not
compatible, the function will set @c out_install_status to
#AR_INSTALL_STATUS_INSTALL_REQUESTED and return immediately. Your current
activity will then pause while the user is offered the opportunity to
install it."

Filing a bug as a part of CR feedback.