If you assume a typical one-week itinerary for your average family,
what are some destinations that are relatively inexpensive once you're
there, based on all-in costs? Maybe I can reconsider some places that
cost an extra $300/per in airfare than the "great deal," but then make
up the $1,200 once we're on the ground.
I wont say don’t go someplace only for cost, because I think that’s got to be pretty obvious, but I will say that if cost is a factor and you do your homework you might find places you hadn’t imagined that are as, or more, interesting to you than many common destinations (here is a thread with some ideas that you might want to see if they light your fire: https://community.ricksteves.com/travel-forum/general-europe/back-doors-to-the-back-doors-of-europe )
First, but not at the sake of making tourism difficult, stay outside of the tourist zones as that will significantly cut your lodging expenses and if you do a little research you can usually find better and cheaper restaurants, often just two blocks outside the tourist zones. Avoid generalizations; David has a point in generality and he did single out Budapest as large and inexpensive so he recognizes exceptions (Bucharest is the 9th largest capital city in Europe and Budapest is 14th, out of 44 to 50 depending on the definition); and they are both inexpensive destinations.
Next, the most obvious choice is Portugal or Spain or Eastern Europe and Google “Cheap European Destinations” and you will get a lot of hits, many with emphasis on this year.
So since you mentioned Paris (not the most expensive); compare it with Budapest (not the cheapest).
According to Numbeo.com Consumer Prices in Paris are 52.4% higher than in Budapest, but knowing how much of that 52.4% filters down to tourism would be another whole study and there is one and it’s at budgetyourtrip.com which says that to visit Budapest is little less than half the cost to visit Paris:
Paris: You should plan to spend around $252 per day on your vacation
in Paris, which is the average daily price based on the expenses of
other visitors. Past travelers have spent, on average, $45 on meals
for one day and $23 on local transportation. Also, the average hotel
price in Paris for a couple is $329. So, a trip to Paris for two
people for one week costs on average $3,533.
Budapest: You should plan to spend around $94 per day on your vacation
in Budapest, which is the average daily price based on the expenses of
other visitors. Past travelers have spent, on average, $23 on meals
for one day and $11 on local transportation. Also, the average hotel
price in Budapest for a couple is $103. So, a trip to Budapest for two
people for one week costs on average $1,317.
Your traveling style will have a huge impact on your actual cost, but I think it's fair to assume that the percentage spread remains constant if your travel style remains constant in one location vs. another. So the “cost” isn’t as important a number as the percentage spread.
Not selling Budapest, but there are literally hundreds of similar destinations and one of them might fit your fancy this year when prices are up and money is tight.
I just got back from Bucharest a few weeks ago and the site pretty much nailed the fact that Bucharest and Budapest cost about the same to visit.
Of course, think outside of the cities too because while Budapest and Bucharest are about the same cost; the smaller destinations in Romania appear to be quite a bit cheaper than similar in Hungary; and if you get out of Paris the gap between France as a whole and Budapest also probably narrows.