What order were the Pauline epistles written, according to evangelical critical scholarship? I am unaware whether this is a hotly contested issue or not, but if it is, I would be looking for the majority opinion among evangelical scholars, that is, scholars who affirm Pauline authorship, inerrancy, and the various other doctrines that make a scholar an evangelical.
Both "majority opinion" and "evangelical" (even with the guidelines you supplied) are slippery terms. They'll probably always be in a state of flux, so it's difficult to supply exactly what you've asked for. But don't panic. I think a relatively clear picture can still emerge.

I've decided to give a sampling of three scholars who I think are firmly within the evangelical camp: Dan Wallace, J. Hampton Keathley III, and Luke Timothy Johnson. For Wallace and Keathley I found dates, but for Johnson I was only able to find an order. Note that Johnson groups together 1-2 Corinthians, 1-2 Thessalonians, and the Pastorals as one unit each.

+-----------------+----------+---------+---------+
|                 | Keathley | Wallace | Johnson |
+-----------------+----------+---------+---------+
| Romans          | 57-58    | 56-57   |       4 |
| 1 Corinthians   | 55       | spr 54  |       2 |
| 2 Corinthians   | 56       | fall 55 |       2 |
| Galatians       | 49 or 55 | fall 48 |       3 |
| Ephesians       | 60-61    | c 60    |       8 |
| Philippians     | 60-61    | c 60    |       5 |
| Colossians      | 61       | c 60    |       7 |
| 1 Thessalonians | 51-52    | spr 50  |       1 |
| 2 Thessalonians | 51-52    | sum 50  |       1 |
| 1 Timothy       | 63-66    | 63      |       9 |
| 2 Timothy       | 67       | 64      |       9 |
| Titus           | 62-67    | 63      |       9 |
| Philemon        | 61       | c 60    |       6 |
+-----------------+----------+---------+---------+


Table formatting provided by Senseful Solutions.

If we consider the same units Johnson does and add the Prison Letters (Ephesians, Philippians, Colossians, Philemon), then we have exactly 6 "units" or "chunks" of letters that need ordering: Romans, 1-2 Corinthians, Galatians, Prison Letters, 1-2 Thessalonians, Pastorals. Using that schema, here's the above table simplified:

+----------------+----------+---------+---------+
|                | Keathley | Wallace | Johnson |
+----------------+----------+---------+---------+
| Romans         | 3        |       4 |       4 |
| Corinthians    | 4        |       3 |       2 |
| Galatians      | 1 or 2   |       1 |       3 |
| Prison Letters | 5        |       5 |       5 |
| Thessalonians  | 1 or 2   |       2 |       1 |
| Pastorals      | 6        |       6 |       6 |
+----------------+----------+---------+---------+


From here, as promised, a clear picture emerges. The consensus is that the Thessalonian letters were the first or second written, Galatians was also early, Romans was written third or fourth, the Prison Letters were later, and the Pastorals last (shortly before the end of his life). The only major point of disagreement is where the Corinthian letters fit in, which is a very complex question as you can read in Wallace (look for the phrase "special problem").