You should be fine.

A real Tweed Deluxe power transformer only supplies 1.65A on the 6.3 current and 2A of the 5V (rectifier) current.  That MM you linked to (despite saying its a clone of a '54) states 4A's for each, which would be more than enough to run 6L6GC's in the amp and your rectifier of choice.  You may consider a SS rectifier to get the plate voltages up some to try can make some use of the extra power handling capabilities of 6L6GC's. 

The output transformer is their straight Tweed Deluxe OT, which is fairly small iron, it'll handle the 6L6's fine, but it's going to limit the power from them.  I have one sitting here at my desk, and it makes my original '60 Tweed Deluxe OT pretty much in every dimension.  BTW, it says it will handle 6V6, 6L6 and EL84, not EL34. 

So, there are two other things to consider,  2x6L6GC's will have a different primary impedance, which means a different secondary impedance, with 6L6GC's, the correct speaker load is 4 ohms (8 ohms with 6V6's/EL84's).  I wouldn't worry to much if you have a stock 8-ohm speaker, but a 4-ohm would give you a little more output.

Also, depending on plate voltage (and voltage from plate to cathode), you may have to switch out values in the bias circuit.  IIRC, you should be okay if you use a 5Y3 (buy NOS, new ones don't use the right voltages) for 6V6 and a 5SR4 or SS rectifier for 6L6GC's.  But without being able to check, I make no promises. 

If you want the fuller tone and smoother, less lo-fi breakup of a 6L6GC, you'll get that.

If you are looking for significantly more headroom, it's probably not going to be a big enough difference.  Maybe you'll coax 18 watts instead of 12, but at stage volumes, that's pretty nominal.  You'd have to try and get around 30 watts or so out of the amp to make a real noticeable different in headroom on stage.  And THAT would take bigger iron and higher voltages. 

So, not that I've bewildered you with details, short answer...you are safe to use 6L6's.  I would use them with a 5AR4 or SS rectifier, both so they bias better and you get higher plate voltages for more output.  Just keep an eye on you tubs to make sure they don't seem to be running incredibly hot ("redplating").

The transformer doesn't care if someone uses octals or novals, 6V6GT's and EL84s are very similar tubes as far as the OT are concerned, same output, same impedance, etc. 

For a PT; a pair of 6V6GT's draw about 1A, a pair of EL84's draws about 1.5A' a pair of 6L6GC's draws about 2A, and a pair of EL34's draws about 3A. 

If you really only want a bit more clean volume, you may get it, but the reality is, most people need enough extra clean volume to get a noticeable difference, and that you may not get.  Just remember, it's only a 12-watt amp and just about everything about its design works against clean headroom, from it's tone stack to its PI, to its voltages, nothing in the amp is headroom friendly.

It may still be full out at 2, you are changing the output section, not the preamp or PI.  Do you have a 12AY7 at V1?  That is the stock 5E3 spec.  People have experimented with various tubes in V1 and V2 to try and control the volume taper.  Preamp tubes won't have much of an affect on overall output volume, but they can offer a little more control over the taper of the Volume pot.