I have a zero based on one type and grain of bullet.

What can I do to calculate what the drop would be if I switch to another bullet, without re-zero'ing?  Is this a simple process, or what calculator can I use with the manufacture's given muzzle velocity, weight and ballistic co-efficient?

I might be over-complicating this, but all the typical drop calculators I have seen or used calculate on a range based zero for the bullet being considered.  Differently designed bullets are going to have different trajectory profiles.  Switching to a heavier bullet will required a different elevation to meet the first bullet's zero.

If you are trying to figure this out to make scope adjustments at distance without resetting your zero on your rifle, a ballistic calculator will be able to do it. I don't know of any calculator that will tell you where a bullet will impact from a rifle that isn't zeroed.
If you are using a ballistic calculator, you can build a new bullet with all of the rifle information. There should be a place to enter zero offset information. Shoot the new bullet at the distance you have the other bullet set up, not changing your zero at all. Just Hold center of target and take note of where the new bullet hits. Enter this info and as long as you've entered your information correctly, the calculator will give you the corrected numbers.

I am not using a scope, just a red dot.  

My intent is get a better appreciation of what will happen when I switch ammo, or to anticipate what the trajectory would be so I can decide if I want to stock a different types of ammo for whatever reason without physically zeroing it.  But it seems that the zero won't make much a a difference if the zero is at a 50 yard distance.

In that case I would just shoot multiple types of ammo and either find ones that have the same or close to the same point of impact, or just make note of the different points of impact.

I have a 69 grain hand load recipe that is the exact same point of impact at 100 yards and is within an inch at 200 in a rifle zeroed for xm193

To the OP: In theory yes, but in practice not necessarily.  Recoil and barrel harmonics can cause different loads to launch at different angels. The only way to really be sure is to put rounds on paper.  If you study the below thread, there's some data from one shooter (me) on this very topic.  In this example, among other things, I tested a Steyr AUG using 55 gr ball, and 75 gr match rounds.  At 200 yards, they had practically the same zero (the zero I used).  And at 50 yards with 55 ball, the 200 yard zero was also spot on for a 50 yard zero (more or less, and this was fairly expected - why I'm a fan of 50 yard zeros).  but the 75 gr rounds were hitting high at 50 yards compared to the 55 ball.  Why?  My theory is the heavier bullets tend to recoil more, and as such tend to leave the rifle at a higher angle than the lighter bullets.  That's not just my theory, Hatcher said basically the same thing in Hatcher's Notebook.

Anyway, here's a link to a thread that includes discussion on this:

So I got an AUG: (including data on impacts of different loads)

In 3 gun, I zero 200 yards for my 77gr home rolled match loads for stages with targets beyond 100 yards. I shoot 55gr bulk for everything else. Inside 100 yards, there isn't much difference. However, after zero for 77gr, I shoot 55gr bulk at 2 known distances and record it. This gives me enough info to adjust POA for the different POI of the two loads.