I managed to figure out a way to achieve what I'm looking for by finding the necessary variable set through minicart.js and then manipulating it via jQuery.
minicart.js gives me the following variable which outputs the current cart total:
paypal.minicart.cart.total();
I then took that variable and applied some jQuery that converted this variable into a separate variable called "cartTotal" which is checked and updated every time a user interaction happens on the page (ie - a click or a keypress). I found that a brief delay was needed in order for everything to work properly. Interestingly, the keypress check needs a longer delay than the mouse click.
My final (for now) working code is as follows:
$( document ).ready(function() {
var cartTotal = paypal.minicart.cart.total();
$('input#cart').val('$' + cartTotal);
$( "body" ).click(function() {
setTimeout(function() { // Setting slight delay on function to accomodate for button push of removing items from cart
var cartTotal = paypal.minicart.cart.total();
$('input#cart').val('$' + cartTotal);
}, 100);
});
$( "body" ).keypress(function() {
setTimeout(function() { // Setting longer delay on function to accomodate for key push of removing items from cart
var cartTotal = paypal.minicart.cart.total();
$('input#cart').val('$' + cartTotal);
}, 900);
});