I understand what you are trying to do and know that this is achievable through custom stylesheets as well alternatively via a custom SSRS Toolbar UI (via old 2005/2010 mgmt and Execution SOAP SSRS APIs*).
But that may be way overkill for your situation.
My best suggestion with the current out-of-box behavior of SSRS as of MSSQL 2019 (applies to 2016 as well) would be to create a custom link or imageLink inside those subreports that you want to have an SSRS Toolbar for which then either pops up a small new browser window above the main report (or on a new tab) with the subreport- and per default SSRS behavior the toolbar will be there, or, if you want to be explicit you can add the SSRS qs param "&rc:Toolbar=true" to that link.
I have actually used both approaches on projects. It really depends on the requirements, priority and time ya know?
Let me know if I can try to further help in any way if anything I shared was useful.
*here is a similar solution to your kind of problem (the SSRS REST API v1 and v2 can do many things but not as much as the old SOAP APIs could): How to export SSRS 2017 report using REST API