Best way to travel from CDG to Hotel du Champ de Mars

If you can't lighten the luggage, I suggest you take a taxi from the Gare du Nord train station.
From CDG, take the RER to Gare du Nord. Then a taxi.
If you can handle the luggage yourself ok, take the Roissy bus from CDG to Opera, then Metro 8 line to Ecole Militaire stop. The hotel is a five minute walk from there.
The cheapest way is to take the RER into Paris, then Metro with at least one change to Ecole Militaire. There are at least two ways to do this. This way involves hauling that luggage up and down stairs more than once!

Ecole Militaire is indeed the closest Metro station. The problem is that almost all (and the ones you would be using) Metro stations involve stairs, and you would need to make at least one change on the Metro also involving stairs.
When I plug in your path at the Ratp trip planner: http://www.ratp.fr/itineraires/fr/ratp/recherche-avancee
and check the wheel chair symbol to get the easiest route, it recommends taking the RER B to St Michel Notre Dame (elevator to the surface) and walking a few blocks to Cluny Sorbonne and taking the 87 bus to Ecole Militaire. That would be the 9 euro train fare plus a bus ticket for 1.80 euro
Another way would be to take the Roissybus to Opera and then take a taxi from there right to your hotel. The Roissybus costs 11 euro and your luggage is stored under the bus. Taxi, I am guessing would be 10-12 euro.
You say "I" so I assume you are traveling alone. Taxi from the airport would be in the 60 euro range.

Another way, and quite cheap, is bus No 350 which will take you from CDG to Gare de l'Est. Then take a taxi to your hotel. The bus costs about 6 euro and there are stops outside each terminal. Your heavy luggage means the RER or Metro will be a royal pain so make your life easy.

"big, heavy suitcase" and budget are not compatible. Pick one. Sorry, but even on the RER dealing with "big heavy suitcase" is a pain. My last trip there was some type of issue. By the time the train came it was PACKED to the brim. I had to sit with my luggage on my lap from Paris to CDG. I would not even have been able to get on the train with "big heavy suitcase"

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I think the compromise is to take one of the buses from CDG to Pairs and a cab from there. Roissy bus might be the best

http://europeforvisitors.com/paris/articles/paris-cdg-ground-transportation-buses.htm

We took the Roissy Bus from CDG to the opera house area--got a taxi in front of Printemps Dept Store if I remember correctly. The cost to that hotel was about 15 euros. No stairs were involved--but if you use RER and the escalator is not working (not unusual ) it would not be fun to haul your luggage down the steps.

I will just add this. There is an escalator up from the Ecole Militaire Metro stop to the exit near La Terrasse Restaurant which is the side of the road you want to be on. Sometimes it is working, sometimes it is not, so plan on having to carry your suitcase up 2 flights of stairs.

I have also stayed at the Hotel du Champ de Mars which I like very much BUT the elevator is tiny. It is one of the ones where 2 people barely fit. You may be able to put your suitcase in it and sprint up the stairs to your level to pull your bag off. However, if you have a really big suitcase it may not fit.

A friend and I also did the trip from Ecole Militaire to CDG via Metro and RER to meet another friend. We didn't have luggage but there was a ton of walking at the station when changing from the Metro to the RER and we decided we would always opt for a taxi to/from that location.

I respect that you have a budget but perhaps you can cut something else to be able to take a taxi to the CdM.

As of March 1, the maximum rate for taxis from CDG to the left bank (where you're staying) is 55 euros.

Thanks everyone for your responses.
Kind regards..
JS

Big suitcase gets you noticed by pickpockets, better to take Taxi for that reason too. As Kim says, Left Bank standard fare is 55 Euro, Right Bank 50. Not to be overly paranoid (and I don't know your arrival time) but it's a potential factor, and as RS is fond of pointing out, the start and end of a trip can be a good time to spend for a Taxi.

edit: re-reading Sam's post the Roissy Bus/Taxi combo is a good one and gets you to the door securely and cheaper.

I was in Paris in December traveling by myself. The RER train from CDG wasn't packed at the airport, but by the time I arrived at my stop it was PACKED. I only had a backpack and 20" roller bag, and it was difficult pushing my way off the train at my stop. Then I walked about 15 minutes through the underground metro tunnels before getting to my "exit" up a flight of stairs to street level. While the metro is great, there are a lot of stairs to go up, down, across, back down, etc. There are a few escalators here and there, but even then, they were either not working or going the wrong direction. With a big suitcase, avoid the metro.