We have baptisms for:

Cape Town, 1692, den 7 September
Een kint gedoopt waer van de vader is Ba....(end of page)
Burger de moeder martha Willemsn/dr
getuygen (.?.) klaasen, ende genaamt Barent

Drakenstein; Le 8 Julliet 1701
Ellaina fille de beerne beurger et de Martha
Eskal Le temoins et françois du toit et Susanne
Seugnet pour marraine.

Drakenstein; 1704, 13 April 
Jacobus
de Vader Barent Burger, de Moeder Marrijtie Willemz, 
getuijgen Martinus Van Stade, Aaltie Willemz.

There are no baptisms of any other children of this couple to be found. I would assume that they were all baptised at Drakenstein where the Registers before 1717 were already known to be missing by 1726 and were later partly reconstructed from the notes of Pastor Simond and others.

Barend Burger died circa 1705. The inventory of his estate (MOOC 8/2, no.3) mentions his widow and his six minor children but does not name them.

There are no Weeskamer records extant for this period (Andre Kellerman investigated).

The Estate distribution accounts¹ for their mother, Marritje Willems Schalck, once again mention six minor children without any names. Her second husband, Lambert Smit, is mentioned in these accounts as the step-father of these children ('haer schoonvader Lambert Smit') and one small note mentions the names Jacobus and Helena Burgert without mentioning who they are, date 17 July 1710:

(Aan) Lambert Smit in voldoening van desselfs geheele 
Erfportie uijt deesen boedel behalven seekere 
f300:-:- die voor een jaer in sequestratie blijft 
tot betalingh van de nog opdagende schulden, en 
om deselve daer na of wel het Restant van dien 
goetgedaente worden aen de Reeken. van Jacobus 
en Helena Burgert jeder voor de helft (f.) 1660,-[]

The earliest date in this inventory is February 1710 but she seems to have died prior to October 1709, since there is an entry in the Estate Accounts of Johannes Bocklenbergh, 
dated 15 Ocober 1709: 

(pr.) den boedel van Lambert Smit en Marritje 
Willems Schalk over verdiend meesterloon en 
geleeverde medicijnen 60,,-,,-

The inventory of the estate of Barend Burger ², drawn up after his death in 1729, states:

So ... deselve bevorens  bij goede mannen en naaste  Bloedvrinden volgens inventaris  sijn opgenomen ... and the inventory itself is signed: In't Land Van Waveren  den 26 May 1729  Willem Burger  franssos dutoij de ijonge josua joubert jacob teron

This  is the weakest point in my argument: 

I am going to assume that Willem Burger was the 'Bloedvriend' (next-of-kin) and the other three were the 'goede mannen' (upright men) and from that I shall draw the very tenuous conclusion that he was a brother of the deceased Barend Burger, one of the three known children of Barend Burger and Marritje Willems Schalck. 

As some slight corroboration, this Willem's first son was named Barend (the convention of naming the firstborn son after the paternal grandfather was followed by many Cape families, as it was by many Dutch families).

Having assumed that Willem Burger was one of the children we move on to his own Estate Inventory³, drawn up in 1731, when Andries Burger attended to oversee the interests of the children, and Jacobus van der Merwe attended on behalf of the widow. 

When Maria van der Merwe, first wife of Andries Burger, died somewhere between 1725 and 1727, an inventory of her and Andries's joint estate was drawn up by Schalk Willems van der Merwe and Willem Burger. Later, Andries Burger, in his will dated 1727 named the above pair once again as the guardians of his children, identifying them as his father-in-law and brother⁴ .

4.
Cape Archives, CJ 2604, 21
Will, Andries Burgert, 27 December 1727.

... den testateur als voogden nomineert aanstelt ende versoekte syn schoonvader den heemraad [Willem - crossed out] Schalk Willemsz vander Merwe en sijn broeder den Landbouwer Willem burgert, ...