Will the 45-day visa exemption allow me to stay in Thailand until the end of April if I arrive on March 15?

you misunderstand how the program works

From Oct 1st until Mar 31 any time you enter the country on a visa exempt entry by air or land you get stamped in for 45 days. EVEN if you entered the country on March 31st you'd get a 45 day entry stamp.

When the program ends has no bearing on your entry stamp so entering march 15 would get you stamped in until about April 28th (44 days from the date you enter because the day you stamp in is day one)