Squad size upgrades are poison, skip all that you are able to.
In a tournament both your squad size and the enemy squad size is a percentage of your squad size. The enemy size ranging from 85% of your squad size for round 1 and 160% of your squad size for round 6. Each round, your own squad size will be 0.05 multiplied by the province number.
So regardless of your squad size, the battles will always essentially be the same difficulty.
Where squad size hammers you is training troops. Imagine you train 1000 troops per hour and have a squad size of 1000 - you train one squad per hour.
Double squad size to 2000 and those it now takes 2 hours to train one squad of those same troops - so it takes you twice as long to train the same number of squads of troops, or you lose twice as many troops per battle, whichever way you want to look at it.
Some numbers if it helps...
Round one of a tourney, in province 1:
You have a squad size of 1000, so based on the information above your squad size for the fights in this province/round will be = 0.05 * 1000 = 50
Your enemy will have a squad size that is 85% of yours = 85% * 50 = 42.
If instead you have a squad size of 2000, your squad size for the fights in this province/round will be = 0.05 * 2000 = 100
Your enemy will have a squad size that is 85% of yours = 85% * 100 = 85.
Round two of a tourney, in province 1:
You have a squad size of 1000, so based on the information above your squad size for the fights in this province/round will be = 0.05 * 1000 = 50
Your enemy will have a squad size that is 100% of yours = 100% * 50 = 50.
If instead you have a squad size of 2000, your squad size for the fights in this province/round will be = 0.05 * 2000 = 100
Your enemy will have a squad size that is 100% of yours = 100% * 100 = 100.
So 50 vs 50 or 100 vs 100, the difficulty is the same - you just need twice as many troops and a higher squad size doesn't help you train troops faster, so you're worse off in terms of how much you can fight before you'll run out of troops (yes, some ancient wonders that produce free troops scale from squad size, but this comes nowhere near offsetting your extra losses from a larger squad size).
Round six of a tourney, in province 1:
You have a squad size of 1000, so based on the information above your squad size for the fights in this province/round will be = 0.05 * 1000 = 50
Your enemy will have a squad size that is 160% of yours = 160% * 50 = 80.
If instead you have a squad size of 2000, your squad size for the fights in this province/round will be = 0.05 * 2000 = 100
Your enemy will have a squad size that is 160% of yours = 160% * 100 = 160.
50 vs 80 or 100 vs 160, you're at the same percentage disadvantage, you'll just lose more troops from the higher squad size, the fight won't be any harder to win.
Squad size upgrades do give you an advantage when fighting enemies in provinces that you've scouted on the world map. However, if you're playing the game in a way to maximise your progress you're going to be doing waaay more tournament encounters than world map encounters. As you're able to go further in the tournament as you grow (do more tournament provinces), the tournament encounters will start to cost much more than the world map ones too.
If you cater instead of fighting, the same applies, the amount of goods/supplies/whatever needed to cater will scale with your squad size, costing you more to cater the same fight than if you had a smaller squad size.
*** When I talk about enemy squad size, the number shown in game might not always correspond per the formulas given. This is where the enemy has more or less than 5 slots. The percentage of the enemy's total size will always be based on the formula divided by the number of slots they have. E.g. Round two of a tourney in province 20, your squad size is 1000, with your 5 slots all counter, your total team squad size is 5000. If the enemy has 5 unit slots in their team the game will show enemy squad size as 1000 too. If the enemy has 4 unit slots in their team it'll show 1250 as their squad size (1250 * 4 = 5000, still the same total as you). If they have 8 unit slots in their team the squad size will show as 625 (625 * 8 = 5000).