Pattern fails before penetration and in a 70mm chambered 20 bore you've only got 28 grams (1 ounce) to play with. You say that you are likely to encounter "partridge, pigeon, duck, pheasants, rabbits" at "expected ranges up to 35ish yds" and that this may be walked up. And you are not expecting "high fast pheasants" so I assume any pheasants will also be inside your thirty-five yard distance.
A pigeon at thirty-five yards is a small target as is a partridge. Therefore although a #5 might kill if it hits a vital spot there may not be enough pellets in your charge of shot to give a pattern that...at that range...can be guaranteed to find that vital spot. But you don't say what choke you are using. Or if you can use plastic wads.
So my advice is the old traditional choke combination where the second barrel is slightly tighter than the first barrel of the old standby of 1/4 and 1/2 and 28 grams of #6 of good quality lead shot...copper plated if you can get it...if using fibre wads. If using plastic wads you could go IMPROVED and 1/4 as a plastic wad gives you the equivalent of an extra degree of choke.
A 28 gram load of #6 will give pattern, more than adequate penetration and power out indeed to forty yards, and should also be easy to find at most gunshops. If OTOH the choice is a "take it or leave it" of 28 gram of #5 or #7 and no availability of #6 I would go for #7 and slightly more open choking of IMPROVED and 1/4.
My late father shot his G E Lewis 20 bore 26" barrel "Light Magnum"
* side by side in the 1960s when lead shot was not as good a quality as today and plated shot, or antimony hardened shot, was commonplace in the usual Eley 20 bore 13/16 ounce 2 1/2" length or Eley 20 bore Alphamax 1 ounce 2 3/4" length so used #5. I think that shot is so much better today...or one hopes that it is if you buy good quality cartridges...that #6 or #7 are now better advised.
* The Lewis "Light Magnum" guns were 262 barrel side by side 20 bore (or also 16 bore) guns that were chambered for 2 3/4" cartridges. Lewis (I met the son who was the "Son" or G E Lewis & Son) when he was then an old man.
The idea of the Light Magnum was a 20 bore gun that fired the equivalent of a standard 16 bore load of 1 ounce and a 16 bore gun that fired the equivalent of a standard 12 bore load of 1 1/8 ounce. The idea was you carried a gun of slightly heavier weight than a standard 20 bore or standard 16 bore but that could hit one level up from that weight.
Here's a link to one of the Lewis 16 bore Light Magnum guns sold at auction last week. They were handsome guns. Very handsome.
bid.harperfield.co.uk