> however numbering the bond 1 means there's ambiguity between it being between 1 and 2 and between 1 and 6

It might *appear* to you all of a sudden that there's an ambiguity, but actually there's none. Look at these two molecules carefully:

[![enter image description here][1]][1]

Note that the numbering for the molecule on the **left** will increase in the **anticlockwise** direction, while the numbering for the molecule on the **right** will increase in the **clockwise** direction. The **pi bond in each case is named as a -1-en** (and it should, because both the molecules are identical)

You seem to be assuming that the numbering for the molecule on the left *also* applies on the molecule on the right. But, that is not the case, as shown above.

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The disubstituted cyclohexene ring complicates the scenario, but only slightly. Consider the case when functional groups of equal priority are present:

[![enter image description here][2]][2]

Now, recognize that - even before assigning a name - that both the molecules are *different*. Thus, their names will also be *different*. The molecule on the left begins numbering from the top $-\ce{COOH}$ and increases anticlockwise. The molecule on the right begins numbering from the lower $-\ce{COOH}$ and increases clockwise. The first pi bond is assigned a name "-1-en" while the other is assigned "-3-en".

If a lower priority functional group is present instead, like so:

[![enter image description here][3]][3]

the numbering will increase *along the direction of the pi bond*, beginning from the top $-\ce{COOH}$. Hence, their names will be 3-hydroxycyclohex-1-ene-1-carboxylic acid and 5-hydroxycyclohex-1-ene-1-carboxylic acid respectively.

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A possible way to achieve a "-6-en" is to take equal priority suffix groups on positions 1, 2 and 4, and then place a pi bond between positions 1 and 6, like so:

[![enter image description here][4]][4]

It is named "cyclohex-6-ene-1,2,4-tricarboxylic acid" giving the suffix functional groups a lowest locant set of "1,2,4", consequently naming the alkene as a "-6-en". Actually, though, this is **not 100% correct**. A slightly technical detail is missing, which can be found in [Loong's answer][5].


  [1]: https://i.sstatic.net/FkTSo.png
  [2]: https://i.sstatic.net/kQzsP.png
  [3]: https://i.sstatic.net/mrgEq.png
  [4]: https://i.sstatic.net/FOPjK.png
  [5]: https://chemistry.stackexchange.com/a/91331/5026