There's no such thing as reliable tanking, until you get enough karma to buy the True Pro upgrade (start with a 2 nuyen card, 30 karma). You're too dependent on the random draw. Big Plans (5 karma) can do it, but it relies on buying a 6+ cost card.
It seems a lot harder for 1 starting card or 1 nuyen to make as much a difference in the game as 1 HP, especially with the first level in a lot of obstacles being multiple damage. There are too many obstacles that do 2 damage and too many ways for a character to end up facing 2 obstacles to be playing with a character that starts with only 4 HP. Alternatively, maybe consider the race cards as a way to set difficulty: Hard Mode is all Elves, Easy(easier) Mode is all Trolls. Since healing can be hard to come by, it's all about starting HP. The exception is that if you don't start with any cards in hand, you just take damage your first turn.
What are the first upgrades I buy? Got Your Backs (pay 3 nuyen to heal a staggered runner 1 HP) and Just Tough (+1 HP), each 5 karma (+10 for the second slot). True Pro (at 30 karma) is the first chance you get to choose a card for your starting deck beside the basics--and the 2 karma cards are decent, though the red one is a lot less frequently useful. Prep Work (10 karma) is potentially useful, but depends on the initial Black Market draw.
If the Crossfire scenario is giving you trouble, try the Extraction scenario. At least you can run out the clock on the Extraction scenario, which can give some leeway when a bad event card comes up. I haven't seen the Harlequin's Shadow card, so I don't know if it makes Close the Portal any less impossible (seems like you have maybe 2-3 rounds before the portal pops and drowns the team in obstacles).