Back in 2018 my rear panel was ordered in December 2017 and arrived for fitting at the Swift dealership in May 2018.The choice is yours. If the other insurer confirms the will meet the repair costs based on the estimate, then they are obliged to do so meaning there should be no risk there. If further damage is found after stripping both insurers would want to check that so again it should not provide a problem. I suspect delays in getting the parts could be the main one but you are approaching a quieter period for the workshops so they might be more keen to do the work at the moment.