Admin Tools does not block you from installing extensions. Moreover, Joomla! itself in its very helpful error message that you pasted above tells you exactly what's wrong. In fact you can go to Extensions, Manage, Warnings and see further information from Joomla! itself regarding this issue.
EXTREMELY IMPORTANT: The php.ini file ONLY applies to the directory it is in. The php.ini file in your site's root DOES NOT apply to the back-end of your site (administrator). You MUST place a php.ini file in the administrator directory as well. Moreover, check with your host whether they allow you to override BOTH the upload size AND the post size.
I'm closing this ticket since it has nothing to do with our software. Please consult with your host about server configuration.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!