A 2 year old needs some space to move around. And a kitchen is definitely a plus when you have a toddler.
You're booking kind of late at this point for an early June stay, and you are also not going to be in Paris for very long (many apartment rental agencies only deal with those who stay for a minimum 5 to 7 days).
Sure, you can try an apartment rental agency, or you could consider upping your budget and looking at some of the apart'hotels, like Citadines, and get yourself a studio or 1 bedroom unit with a cot or pull-out sofa for the kid. But in your price range, I'd suggest you consider looking at staying in one of the family rooms at a hostel. Benefits are you have not only access to a kitchen to eat breakfast, pack a lunch or make a simple dinner for your child at whatever time he/she needs to be fed, but a hostel also will give you access to common area lounges with plenty of room for your kid to move around. You also will find most hostels provide some sort of coin-op washer/dryers on site (helpful with a small child).
Read reviews of any hostel you might consider on sites like hostelworld.com and see what others have to say about the rooms, location, amenities and the overall atmosphere. Steer clear of the ones where reviewers rave about partying. Know that some hostels (particularly those in the Hosteling International chain) tend to attract lots of school groups so again, see what reviewers have to say. But many families love booking a private room in a hostel - and hostels are more frequently squeezing in en-suite bathrooms inside rooms wherever they can. Even if a family room is booked, some hostels have triples or quad dorm rooms with their own private bathroom. If you happen to find a quad that looks like it would work, you can always just pay the cost of the extra bed you won't need just to ensure you have the entire room all to yourself. You'd likely still be well within your nightly budget and you could probably stay somewhere a little more central rather than commute a longer distance.
If you still prefer a hotel, check Accor hotels (Ibis, Mercure, etc.) to see what options they have in your desired price range.
