The best way that I've seen to handle this is to draw the map in chunks. Before the session, draw the first several rooms of the crawl, either as part of your pre-game prep or as people are coming in and getting set up.
If you have a break in your session (say, to order pizza), spend some time during that break erasing the map and drawing out a new set of chunks based on the players' location.
Most of the time the layout of the map isn't actually important (since the PCs can only really see the shape of the room). If it is important, just cover the map with paper, and remove sections that covered areas explored by the PCs.
You can also print a copy of the map without details on it, and hand that to a PC. Have them draw the map as you're doing the initial narrative.
This will work well for tactical games, like D&D 3 or 4, where the players move through the dungeon relatively slowly (room to room). For games where the players might fly through rooms quickly, use a small-scale map for the overall dungeon layout, and use the battle mat for the rooms where the fighting actually takes place.
The DMG rules are consistent if you count the exact number of spaces, while measuring either way around.
In other words, if you have two creatures adjacent to a middle creature, then count both clockwise and counterclockwise. If the number of spaces (in either direction) equals the number you're trying to count, then there are that many spaces in between.
The trouble here is a geometric property that on a hex grid, Large and Gargantuan creatures are surrounded by an odd number of hexes. If you pick two of these hexes (as possible flanking locations), and count the spaces in between, then you are guaranteed to get different measurements clockwise versus counterclockwise.
However, it is possible to pick two surrounding hexes such that, if you count the in-between spaces both clockwise and counterclockwise, the measurements differ by 1. This is the closest way to split the difference (an odd number), so the DMG uses these measurements to determine flanking. Thus if a creature is surrounded by an odd number of hexes (i.e., they are Large or Gargantuan), then an adjacent attacker can flank with up to two other adjacent attackers.
When two Medium creatures are adjacent to a Gargantuan creature, there are 13 free adjacent spaces. If you can count 6 one way, then you must count 7 the other way. In this case, you counted the short way, but you still counted exactly 6, so they are flanking.
When two Medium creatures are adjacent to a Large creature, there are 7 free adjacent spaces. If you can count 4 one way, then you must count 3 the other way. In this case, you counted the long way, but you still counted exactly 4, so they are flanking.
By comparison, with Medium and Huge creatures, they are surrounded by an even number of hexes. It is possible to pick two of these hexes, such that you can count the same number of spaces in between both clockwise and counterclockwise; these are visibly the flanking positions.
As a side note, the second DMG quote only applies to a square grid. Otherwise you are dealing with creatures that don't have "opposite sides or corners".
So for the purpose of determining whether two attackers flank a Large (or Gargantuan) creature, you will have to count either clockwise or counterclockwise.
In the above examples, there are exactly 4 spaces counted in between the attacking creatures.
Best Answer
D&D 5e does not need a battlemap ... but you knew that already, right?
However, to take it further, a bettlemap adds to D&D 5e for close range (melee) combat but adds next to nothing for long range combat. Unless and until the combat reaches a range where melee combat is a possibility, just do without the map until that happens.
Instead, keep track of how far the groups are from each other until the range closes. You can do this precisely if you like or, for normal movement creatures (20-40 feet) if you just keep track of how far apart they are in 30 foot increments and each time a group uses movement (or Dashes) to close the distance, reduce the range by 30 feet. In general, most groups will want to stay in support distance of each other so they will move at the speed of their slowest member. If some do want to dash ahead or stay behind, just break up the group into as many as you need.
If you like, you can get each side to set themselves up 6 squares apart in the relative positions they want and write the range between them, when it reaches 0 they are in melee range and use the map normally.