CH32V003J4G6

Did you make a typo? There's no such part as the J4
G6. I assume you mean the 8-pin CH32V003J4M6.
According to the datasheet, PC has the alternate function of TIM2CH4_.
I assume I am missing some sort of mapping config function call needed to have PC1 output TIM2CH4 there. I am confused on the meaning of the '_' designation of TIM2CH4_ here and what is needed for it.
Take a look at section 7.2.11 of the Reference Manual. Table 7-9 shows what AFIO register settings need to be made to map TIM2CH4 to different pins.
The pin labelling convention is a little confusing at first glance, but makes sense. On the Datasheet pinout the trailing underscore just means it is one of the alternate mappings. You need to refer to Datasheet table 2-2 for all the possible alternate function re-mappings. There, "T2CH4" (no underscore) is the default mapping on PD7, "T2CH4_1" is the first alternate mapping on PC1, "T2CH4_2" is the second alternate mapping on PD7 (yes, that's the same as the first... I'm sure it makes sense to someone

), and "T2CH4_3" is the third alternate mapping on PD5.
Of course, the 8-pin package has no PD7, only PD5 and PC1, so you definitely need to remap TIM2CH4. Beware that PD5 is physically commoned with PD1 and PD4 on the 8-pin chip, so if using PD5 you need to ensure the other two are configured as floating inputs.
You want to set the
AFIO->PCFR1 register's
TIM2_RM field to 0b01 to remap TIM2CH4 to pin PC1. I'm not exactly sure what you need to do with the HAL API to do that. Looks like you need to call
GPIO_PinRemapConfig(GPIO_PartialRemap1_TIM2, ENABLE). Don't forget to also enable the peripheral clock for AFIO.