Here's the thing.  I could do the work for you if you wanted to put it out.  However, I absolutely cannot front the costs ahead of time, as I don't have a day job and do not have several hundred dollars to buy PCBs, shells, etc.  But if you're willing to front the costs to have the parts shipped to me, I can assemble the carts for a to-be-determined fee.  Additionally, I can print the labels for them myself (I already do that for the NES repros I make).  I can even print boxes off for them if you would like.  Probably manuals too, but I haven't found a paper type that I like for it.  For the overlays I'm thinking I can print on card stock and laminate them with the same stuff I use to make my labels.

Now, the hard part to hear would be the project viability from the financial aspect.  I'm assuming this won't be more than a 4K game.  I found boards that run about $5 each, with discounts for larger quantities.  A quick look for the chips...I found a supplier selling at about $1/chip for 2K or a bit over $2/chip for 4K.  And that's assuming those suppliers have the desired quantity available, but those are pretty reasonable (if low) estimates.  Then you have to buy cart shells, which I'm finding hard to source but could be asked around for.  I'm assuming based on NES that you'll be looking at $5-10 per shell, but that's purely guesswork.  I do have a friend with a 3D printer that might be willing to do it (he's just upgraded to a $2000 printer so he may be keen to try it out), but they're not generally as nice as the injection moulded ones.  Anyway, based on just that, we're looking at up to $22 in cost per cart, before we factor in my fee for assembly, testing, and labels.  Note - bulk buying may lower that cost substantially, but that won't be known until you're certain it's worth doing and start digging into it yourself.  And if you want to add extra stuff, like the overlays, and potentially manuals and/or boxes, that just adds to the per unit cost.

So what you need to do is figure out two things.  How much you think someone would buy it for, and how many copies you think will sell.  So let's say you think 20 copies will sell, and would sell for $30 loose w/overlay.  We'll use the high end and say $22 per for parts.  I usually do $5 for a label on it's own, but for 20 of them I'll be generous and say $3 each.  Additionally, my time would need to be factored in.  I'm going to assume it'll be about 8 hours of work total (this includes printing the labels and overlays, cutting everything to size, and the materials that I use to do that side, in addition to the assembly costs).  I'm again gonna be generous and say $100 in labour costs (works out to about $12.50/hr) - note this is NOT a quote, just a hypothetical...a quote will be given with real numbers when/if we do this.  So we're looking at a cost of $25*20 + $100, or $600 total, requiring a per unit cost of $30 just to recoup the cost.  So your net profit will be zero, and that doesn't factor in the cost of me sending you the inventory in order for you to actually sell them. 

To be clear, I am not in any way trying to discourage you from making this, and honestly I would love to work out a deal so we can both make some money and get your game to market.  I just want to make you aware of the fact that you could be out hundreds of dollars up front with no way to recoup the money at the end of it.

What I would suggest is, rather than publishing this particular game (unless you think it's a guaranteed blockbuster), just use this as a stepping stone to get experience programming, and work on a bigger project that would be a guaranteed must-buy.  I would gladly collaborate with you to make your dream a reality, but I don't want you to pull an NGD and make 250 copies of a game that ends up selling 50