I've finally got an answer from Sitecore Support - unfortunately there is no way to send POST requests to Solr (7.2.1) from Sitecore (at least 9+). Hope it will be fixed soon, but as of now - there is an official response:
Thank you for contacting our support team. From the provided description, it looks like you've faced a bug with reference number 288837, because of which setting "ContentSearch.Solr.SendPostRequests" does not work with value "true". More information about public reference numbers can be found here: https://kb.sitecore.net/articles/853187
According to our tickets history, changing the "ContentSearch.Solr.SendPostRequests" value to "false" and increasing a header size as per the mentioned by you KB article is the only known workaround at the moment.