Back when the EU was small it's easy to see at a glance that the opposite was the case, i.e. the small countries were more satisfied with membership. Germany, France, and the UK were all below the Union-wide satisfaction average in 2002, although Germany was closest to the average. With the exception of Italy and Spain the most satisfied countries were all small. However even back then there were some discontents among the smaller countries as well: Finland, Austria, and Sweden.

Here's the raw data on same question asked more recently.

There are some large changes relative to the 2002 poll, e.g. Italy stands out among the miscontents now and Germany is now way above average in terms of membership-satisfaction. I've also noticed they've asked a new question now, whether people think their voice counts. I can already tell that some small countries (e.g. Estonia) are both happy being in the EU and they think that their voice doesn't count!

Just using the membership...good-thing data, the answer is still no; there's still a slight downward trend for the larger countries.

(Population data for 2017 from Wikipedia, in millions.)
The small countries are all over the place in terms of membership-satisfaction.
Since the population isn't normally distributed by country, here's the same graph with log(pop) axis, so it's easier to see the small countries
:
Here's how make your hypothesis a little more true: Brexit

However, in log(pop) terms there's no change in the sign of the slope after Brexit (graph omitted).
Also, since the 95% confidence zone (dark grey area) can swing both ways in all the linear pop graphs, no real conclusion can be drawn, other than the fact that the vagaries of the satisfaction in the few large countries largely influence the (upward or downward) slope of the trendline. The [lack of] confidence observation even holds for the log(pop) graphs.
If somebody wants the raw data to play some more, here it is:
country sat pop
1 LU 85 0.591
2 IE 81 4.784
3 DE 79 82.522
4 NL 79 17.082
5 DK 76 5.749
6 MT 74 0.460
7 PL 70 37.973
8 EE 69 1.316
9 ES 68 46.529
10 SE 68 9.995
11 BE 67 11.352
12 LT 67 2.848
13 PT 65 10.310
14 FI 61 5.503
15 HU 61 9.798
16 RO 59 19.644
17 SI 58 2.066
18 FR 55 66.989
19 BG 54 7.102
20 LV 52 1.950
21 CY 52 0.855
22 SK 50 5.435
23 UK 47 65.808
24 AT 45 8.773
25 EL 45 10.768
26 IT 39 60.589
27 HR 36 4.154
28 CZ 34 10.579