Approximate time to get from Gare Montparnasse to Gare Lyon at 3pm

I'm going to assume you are referring to the train stations, Gare Montparnasse and Gare de Lyon. Yes, barring disruptions or unplanned delays/stoppage it is possible. Use the RATP website for options travel options within Paris, but metro or taxi is a coin toss as to transfer times. Use the SNCF website to check for train times. Of course you also need to allow time to buy your tickets, and also to get from Fontainebleau station to the rental agency.

You are going to the Gare de Lyon, not the city of Lyon.
The fastest way between the two stations is bus #91 which has a dedicated bus lane or taxi which can use the bus lane.
Your timing is close so no one can guarantee you’ll make it to a car rental by 5:45.
If the train is on time, it will take
15-30 minutes to get to the bus stop or taxi stand.
Wait for bus or taxi.
20-30 minutes to Gare de Lyon.
Wait for the next train
One hour travel to Fontainebleau.
Very close timing.

Thank you for the responses. I assumed i have some room to spare and was just being cautious, so I am a bit surprised that the feedback is that i will be cutting it close. So would it be safer for me to just rent the car the next day, even though it has its own inconveniences?

We visited Fontainebleau this weekend, and our train out from Gare de Lyon was delayed by over half an hour. There were no announcements or anything, so we had to check the twitter feed for updates, and based on other posts it seemed like this was not an uncommon occurrence- so you may want to take that into account. The travel time, when not delayed, is 45 minutes.

I just did your trip using the RATP website for tomorrow at 3:30 PM and it says that the train with a connection will get you from Gare Montparnasse to Gare de Lyon in 33 minutes and the bus will take 41 minutes. The train to Fontainebleau takes about 45 minutes and I am not sure how frequently they run (I'm going to Fontainbleau tomorrow morning) but you have to assume that there will be some waiting at Gare de Lyon for your train. If you caught the 4:32 train, you would arrive in Fontainebleau at around 5:12 so you basically have a 30 minutes of leeway. I probably would not risk it because trains, as mentioned above, can be late etc. Is there a reason why you cannot rent a car and drive from Gare Montparnasse to your B&B instead of trying to get across town to take a second train to rent a car? Have you confirmed that there is a car rental location at the Fontainebleau train station?

JHK, we did consider driving from Paris to Fontainebleau ourselves. but decided against it. Seems like driving in Paris, especially during rush hour, may be stressful. Are we wrong?

There is no car rental location at the Fontainebleau train station, but there is one in downtown.

I don't find driving from Gare Montparnasse to the Périphérique especially challenging but I am not especially worried about driving in Paris and have rented cars and driven into Paris and out of Paris a number of times. But if you do not have a confident driver or are anxious about it or not used to city driving, your concerns could be valid. I lived in New York City and Boston for a total of about 25 years and drive in NYC often so that could be a factor. If you stick with your original plan, will you have luggage? I am assuming that you are arriving from someplace like Toulouse or Bordeaux and so driving directly to Fontainebleau is out of the question. What is your plan for getting from the train station in Fontainebleau to the car rental? By the way, to your original question, if you have a lot of luggage, I'd say take a taxi even though I think it will be slower than the metro or the bus.

Where are you arriving from?

Could you get a train to Massy TGV instead of Montparnasse, and rent the car at Massy TGV? It is an easy drive to Fontainebleau from there; a bit of traffic on the A6 between Savigny and Evry but manageable mid-afternoon.
Edited to add that if you are coming from points south of Tours, such as Bordeaux, another option is taking the train to St Pierre des Corps near Tours and driving the rest of the way: a very easy drive on the A10 and A19 to Pithiviers, then D2152/152 to Fontainebleau.
And if you are starting from the Loire valley, I would just rent a car there and not take the train at all.

If neither is an option, the drive from Montparnasse to Fontaineableau won't be horrible at 3:30. Escape from the Montparnasse area via Place de Catalogne, rue Alain and rue Vercingétorix, and you'll quickly reach Porte de Vanves. From there, either enter the Périphérique (Paris Beltway) or just stay on the frontage road (Boulevard Adolphe Pinard / Romain Rolland - less stressful) until Porte d'Orleans, where you'll see the ramp to A6a towards Lyon-Evry.

And if you are starting from the Loire valley, I would just rent a car there and not take the train at all.

This is the only course of action that makes any sense.

I think the times across Paris at 3 pm are very optimistic. We have had a cab across town at this time of day take over an hour. I am leery of plans that require everything to go right.