Can you log in to your dbms server with some MySQL client, like phpmyadmin? If so you can give these sorts of SQL statements to upgrade your tables.
Always start with the strange-looking SET
statement; it makes sure MySQL won’t choke on the default-0 date columns in some tables when it does formatting or keying operations on the table.
SET @@sql_mode := REPLACE(@@sql_mode, 'NO_ZERO_DATE', '');
ALTER TABLE wp_postmeta ENGINE=InnoDb, ROW_FORMAT=DYNAMIC;
ALTER TABLE wp_posts ENGINE=InnoDb, ROW_FORMAT=DYNAMIC;
This would change those two tables over to InnoDB.
The timeout issue? Doing this kind of thing from phpmyadmin or some other SQL client program means it’s not affected by the web server timeout. The Index WP MySQL For Speed dashboard user interface is, of course, affected by that. But not the wp-cli interface, read on.
If you can use wp-cli, you can install Index WP MySQL for Speed, then use a command like this
wp index-mysql upgrade wp_posts wp_postmeta
to upgrade a couple of tables to InnoDB. This doesn’t time out either. If you want a guide to what that cli can do, say
wp index-mysql help
ALL THAT BEING SAID, you know and I know what serious problems can come from hosting services with years-obsolete software versions. Not only performance and support problems, but also information security problems. I wish you the best of luck persuading your client of this and getting them to move. Because move they should.