Hi,
That log line is the answer we were missing. The install is being aborted by ModSecurity, the web application firewall running at your host's Apache level, not by anything inside Joomla:
ModSecurity: Rule [id "35013"][line "40"] - Execution error - PCRE limits exceeded (-8)
[file ".../modsecurity-crs/activated-rules/modsecurity_00_combell.conf"]
[uri "/administrator/index.php"]
What happens: HikaShop's install sends a large request to /administrator/index.php. One of your host's firewall rules (id 35013, in a Combell ruleset file) tries to scan that request, its regular expression hits the PCRE backtracking limit and errors out (that is the -

, and the error kills the request in the middle of the install. So the component and plugin rows get written to the extensions table, but the CREATE TABLE step never runs, which is exactly the "hikashop_config doesn't exist" symptom.
This also explains everything else you found. It is at the web-server level, so removing your Joomla plugins, reinstalling the core and resetting the .htaccess could not change it, and the clean test site only worked because the same rule did not trip there.
Only your host can change this. Go back to them, this time with the exact rule instead of just "the install fails". The rule file is theirs (modsecurity_00_combell.conf, so this is Combell), and ask them to either:
1. whitelist or disable ModSecurity rule id 35013 for /administrator/index.php on this site, or
2. raise the PCRE limits (SecPcreMatchLimit and SecPcreMatchLimitRecursion) so the rule can finish evaluating the request instead of erroring.
Either one lets the install request through and the tables get created. If your hosting control panel has a ModSecurity on/off switch, you can also disable it just for the install and turn it back on afterwards.
Before the next attempt, delete the leftover HikaShop rows from the extensions table again so you start from a clean state.